\section{Introduction}

\IEEEPARstart{A}{utonomous} vehicle technologies for self-driving have attracted significant public attention through conveying huge benefits such as lower vehicle emissions, less traffic congestion, better fuel economy, or shorter travel time \cite{RN757, skeete2018level}. The current self-driving cars in the transportation system stay at simplified levels and need to consider efficacious and practical issues, including precise controls, costs, liability, privacy, and safety \cite{RN758, shneiderman2020bridging}.

Although traditional control theory and supervised learning applied to path planning of autonomous driving have been investigated since the last century \cite{RN760, bazzan2009opportunities, hussain2018autonomous}, reinforcement learning (RL) has recently demonstrated powerful performance, with a basis of enabling a vehicle to learn optimal control policies by interacting with the environment, without the need for prior knowledge or a large amount of training data \cite{wu2017flow}. Furthermore, deep reinforcement learning (DRL) models have achieved some notable successes \cite{RN676}. For example, AlphaGo \cite{RN674} defeated a human professional in the board game Go in 2015, which was the first time a computer Go program had defeated a professional human player in history. The deep Q network (DQN) \cite{RN673}, an off-policy method that adds a target network for stability and prioritised experience replay for efficient sampling, demonstrated high efficiency in playing 2600 Atari video games and outperformed humans just by learning from raw image pixels. Furthermore, advantage actor-critic (A2C) \cite{mnih2016asynchronous}, as another off-policy method that uses advantage estimates to calculate the value proposition for each action state pair, is a lightweight framework that uses the synchronised gradient to update that keeps the training more cohesive and potentially makes convergence faster. In addition, proximal policy optimisation (PPO) \cite{schulman2017proximal} is an on-policy method that provides better convergence with simplicity to implement and tune in most benchmark RL testing environments and was applied to traffic oscillations \cite{li2021reinforcement} and hierarchical guidance competitions \cite{cao2021reinforcement,cao2021weak}. These DRL research initiatives have stimulated more recent research areas on the application of DRL in autonomous driving and provided promising solutions in predictive perception, path and motion planning, driving strategies, and low-level controller design \cite {RN749}. However, they paid too much attention to evaluating the reward coverage returns and ignored the crucial safety factors in autonomous driving.

\section{Related Work}

The vehicle needs to reinforce learning from experience, usually at the cost of collisions, to gain autonomous knowledge and achieve higher rewards. Recent research on the DRL-based driving domain focused on the long-term accumulative reward or averaged reward as the critical performance metrics \cite{zhou2020smarts, RN655}. Nevertheless, the literature rarely paid attention to a safety measure to evaluate the autonomous performance of DRL models. One recent study \cite{RN775} placed a focal point on the vehicle's safety issue but only applied it to a straight four-lane highway road instead of a road traffic junction driving scenario.

To investigate the self-awareness mechanism for DRL, we paid heed to the self-attention module that was first introduced by \cite{RN654} to resolve the inability to translate complex sentences in natural language processing tasks, which enables the memorisation of long source sentences by introducing a dynamic context vector of importance weights. Afterwards, \cite{RN662} developed the transformer model, which is a multi-head self-attention module and achieved outstanding performance for translation tasks. Furthermore, \cite{RN726} used a variant form of the transformer model to accommodate the varying size of input data from the surrounding vehicles in autonomous driving, particularly in advance of addressing the limitation of the list of feature representations in a function approximation setting and enhancing the interpretability of the interaction patterns between traffic participants. However, the current explorations did not discover an attention mechanism to evaluate the safety concerns of an autonomous vehicle in the driving environment, specifically challenging road traffic junction scenarios.

In this work, we \textbf{aim} to (1) evaluate the safety concerns from four metrics (collision, success, freezing and reward) from three state-of-the-art baseline DRL models (DQN, A2C, and PPO) in autonomous driving, especially in complex road traffic junction driving environments, such as intersection and roundabout scenarios; (2) propose a self-awareness module for the DRL model to improve the safety performance in road traffic junction scenarios. The outcomes of this work will contribute to safe DRL for autonomous driving.

\section{Methodology}

\subsection{Baseline DRL Models}

\subsubsection{DQN}

DQN \cite{RN673} is a relatively typical DRL model inspired by \cite{RN744}, where the goal of the agent is to interact with the environment by selecting actions in a way that maximises cumulative future rewards. DQN approximates the optimal action-value function as follows:

\begin{equation}
{Q^*(s,a)=\mathbb{E}_{s'\sim\epsilon}\left[r+\gamma\max_{a'}Q^*(s',a')|s,a\right]}
\end{equation}
where $Q^*(s, a)$ is the maximum sum of rewards discounted by $\gamma$ at each time step, achievable by following any strategy in a state $s$ and taking action $a$. This function follows the Bellman equation: if the optimal value $Q^*(s', a')$ of the state $s'$ at the next time step was known for all possible actions, then the optimal strategy is to select the action $a'$ maximising the expected value of $r+\gamma Q^*(s', a')$.

DQN has some advancements used for autonomous driving. First, DQN employs a mechanism called experience replay that stores the agent's experience $e_t=(s_t, a_t, r_t, s_{t+1})$ at each time step $t$ in a replay memory $D_t={e_1,\dotsc,e_t}$. During learning, samples of experience $(s, a, r, s')\sim U(D)$ are drawn uniformly at random from the replay memory, which removes the correlations in the observation sequence and smooths over changes in the data distribution. Second, DQN uses an iterative update to adjust the action values Q towards the target values. A neural network function approximator (Q-network) with weights $\theta$ is used to estimate the Q function. The Q-network updates only the target values periodically to reduce correlations with the target, and the Q-network is trained by minimising the following loss function $L$ at iteration $i$:

\begin{equation}
\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
\Scale[0.9]{L_i(\theta_i)=
\mathbb{E}_{(s,a,r,s')\sim U(D)}\left[{\left(r+\gamma\max_{a'}Q(s',a';{\theta_i}^-)-Q(s,a;\theta_i)\right)}^2\right]}
\end{equation}
where $\theta_i$ denotes the parameters of the Q-network at iteration $i$. ${\theta_i}^-$ are the parameters of the target network at iteration $i$, which are held fixed between individual updates and are only updated with the Q-network parameters $\theta_i$ every certain step.

\subsubsection{A2C}
A2C is a synchronous, deterministic variant of asynchronous advantage actor-critic (A3C) \cite{mnih2016asynchronous}. For autonomous driving, A2C maintains a policy $\pi(a_t|s_t;\theta)$ and an estimate of the value function $V(s_t;\theta_v)$, which uses the same mix of $n$-step returns to update the policy and the value function. The policy and the value function are updated every $t_{max}$ actions or when a terminal state is reached. The update performed by A2C can be denoted as $\nabla_{\theta^{\prime}} \log \pi\left(a_{t} s_{t} ; \theta^{\prime}\right) A\left(s_{t}, a_{t} ; \theta, \theta_{v}\right)$, and $A(s_t,a_t;\theta,\theta_v)$ denotes the advantage function as follows:

\begin{equation}
\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
\Scale[1]{A(s_t,a_t;\theta,\theta_v)=\sum_{i=0}^{k-1}\gamma^i r_{t+i}+\gamma^kV(s_{t+k};\theta_v)-V(s_t;\theta_v)}
\end{equation}
where $k$ denotes the number of actions taken since time step $t$ and is upper-bounded by $t_{max}$.

\subsubsection{PPO}
PPO is a model-free, actor-critic, and policy-gradient approach to maintain data efficiency and reliable performance \cite{schulman2017proximal}. In PPO, $\pi$ denotes the policy network optimised with its parameterisation $\theta$, and the policy network takes the state $s$ as the input and outputs an action $a$. PPO uses actor-critic architecture to enable learning of better policies by reformulating reward signals in terms of advantage $A$. The advantage function measures how good an action is compared to the other actions available in the state $s$. PPO maximises the surrogate objective function as follows:

\begin{equation}
\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
\Scale[1]{L(\theta)=\hat{\mathbb{E}}_t[min({r_t(\theta)\hat{A}_t,clip(r_t(\theta),1-\epsilon,1+\epsilon)\hat{A}_t})]}
\end{equation}
where $L(\theta)$ is the policy gradient loss under parameterization $\theta$. $\hat{\mathbb{E}}_t$ denotes the empirically obtained expectation over a finite batch of samples, and $\hat{A}_t$ denotes an estimator of the advantage function at timestep $t$. $\epsilon$ is a hyperparameter for clipping, and $r_t(\theta)$ is the probability ratio formulised as:

\begin{equation}
r_t(\theta)=\frac{\pi_\theta(a_t,s_t)}{\pi_{\theta old}(a_t,s_t)}
\end{equation}

For autonomous driving, to increase sample efficiency, PPO uses importance sampling to obtain the expectation of samples gathered from an old policy $\pi_{\theta old}$ under the new policy $\pi_\theta$. As $\pi_\theta$ is refined, the two policies will diverge, and the estimation variance will increase. Therefore, the old policy is periodically updated to match the new policy. The clipping of the probability ratio to the range of $[1-\epsilon,1+\epsilon]$ ensures that the state transition function is similar between the two policies. The use of clipping discourages extensive policy updates that are outside of the comfortable zone.

\subsection{Self-Awareness Safety DRL}

The intuition of a self-awareness safety mechanism driven by the attention modules enables the ego vehicle to filter information and emphasises relevant vehicles that have potential conflicts with the ego vehicle's planned route, and the ego vehicle will be more informed to make decisions that could avoid collisions to stay safe. Following this motivation, we proposed a self-awareness safety DRL that employed the self-attention architecture \cite{RN726} and connected a normalisation layer to improve the training speed of multi-head attention based on the transformer framework \cite{RN765}. Second, we incorporated the proposed self-attention architecture with a baseline DRL model to build a self-awareness safety DRL, such as attention-DQN, to evaluate the safety performance of the ego vehicle in intersection and roundabout driving scenarios.

\subsubsection{Attention-DQN}

\begin{figure*}[!t]
\centering
\includegraphics[width=13cm]{figures/fig1_DQN_architecture.pdf}
\caption{Our proposed self-awareness safety DRL (attention-DQN). }
\label{fig_DQN_architecture}
\end{figure*}

As shown in \textbf{Figure \ref{fig_DQN_architecture}}, we presented a self-awareness safety DRL: attention-DQN, in a simulated road traffic junction environment including intersection and roundabout scenarios. The ego vehicle is an agent who observes the state $s_t$ of the environment and decides an action $a_t$; for instance, the vehicle accelerates forward. The environment then will give a reward $r_t$ as feedback and enter a new state $s_{t+1}$. A DQN agent contains two deep networks: the Q network and the target network; the Q network includes all updates during the training, while the target network is used to retrieve Q values. The target network parameters are only updated with the Q network parameters every 512 steps to reduce the correlations between Q-value updates and thus make the Q-network output's distribution stationary \cite{RN766}. DQN also uses the concept of experience replay to form a stable enough input dataset for training, i.e., all experiences in the form of $e_t=\left(s_t, a_t, r_t, s_{t+1}\right)$ are stored in the replay memory and sampled uniformly in a random manner.

Furthermore, the two-deep networks in \textbf{Figure \ref{fig_DQN_architecture}} are considered attention networks, and the architecture of an attention network is illustrated in \textbf{Figure \ref{fig_architecture_attention_network}}. For an attention network, the observations containing vehicle features (i.e., location and velocity) are first split into the ego vehicle's inputs and those of other surrounding vehicles. The input observation information is normalised and then passed to a multilayer perceptron (MLP) network, and then the outputs from both MLPs are passed to a multi-head (i.e., two heads) attention block. Specifically, the multi-head attention block contains a single query $Q$ from the ego vehicle, keys of all the vehicles $K$ and values of all the vehicles $V$. Linear transformations are applied to the $Q$, $K$, and $V$, as well as the scale dot-product attention that is calculated from the transformed data. Then, the attention data from multiple heads are concatenated and linearly transformed before being passed to the next layer. Afterwards, the attention data from multiple heads are combined and added to the MLP output of the ego vehicle and then fed to a LayerNorm layer. Finally, the regularised data from the normalisation layer are passed to the output MLP to produce the action values \footnote {Source code is available at \href{https://github.com/czh513/DRL_Self-Awareness-Safety}{GitHub}. }.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/fig2_attention_architecture.pdf}
\caption{The architecture of a self-attention network.}
\label{fig_architecture_attention_network}
\end{figure}

\subsubsection{Default Single-query and Multi-query Attention-DQN}

Here, we introduced \textbf{Figure \ref{fig_attention_head}} to further present how an attention head works. In terms of the single-query attention architecture shown in \textbf{Figure \ref{fig_attention_head}-A}, first, the ego vehicle emits a default single query $Q= [q_0]$ to select a subset of vehicles according to the context. The query is computed with a linear projection $L_q$. Second, the query is compared to a set of keys $K= [k_0, …, k_N ]$ that contain descriptive features $k_i$ for each vehicle, and the keys are computed with a shared linear projection $L_k$. Third, the dot product of the query and keys $QK^T$ is computed to assess the similarities between them. The result is scaled by $( 1 / {\sqrt{d_k}})$, where $d_k$ is the dimension of keys, and a softmax function is applied to obtain the weights on the values $V= [v_0, …, v_N ]$. The values are also computed with a shared linear projection $L_v$. Please note that the keys $K$ and values $V$ are concatenated from all vehicles, whereas the query $Q$ is only produced by the ego vehicle. The output attention matrix is formalised as:

\begin{equation}
\label{eqn_attention}
output=softmax\left(\frac{QK^T}{\sqrt{d_k}}\right)V
\end{equation}

In addition, we explored a multi-query attention architecture, marked as "MultiQ-attention-DQN", as demonstrated in \textbf{Figure \ref{fig_attention_head}-B}, which tweaked from the single-query attention architecture to observe the performance variations, and the main differences were highlighted in orange. Here, we include queries from all the vehicles instead of using the default single query from the ego vehicle.

\begin{figure}[!t]
\centering
\includegraphics[width=7cm]{figures/fig3_attention_head.pdf}
\caption{The architecture of an attention head: (A) single-query attention head; (B) multi-query attention head.}
\label{fig_attention_head}
\end{figure}

\section{Experiments}

\subsection{Driving Environment}

In this study, we used two representative driving scenarios, intersection and roundabout, as shown in \textbf{Figure \ref{fig_scenario_eg}}, to investigate the safety of DRL for road traffic junction environments based on the highway simulation system \cite{RN754}. We suggested intersection (\textbf{Figure \ref{fig_scenario_eg}-A}) and roundabout (\textbf{Figure \ref{fig_scenario_eg}-B}) scenarios characterised as relatively challenging driving interflow environments. It includes one ego vehicle trying to cross the traffic and arrive at a destination, and several surrounding vehicles spawned randomly. The ego vehicle needs to make decisions such as turning left, changing lanes, accelerating and decelerating while avoiding colliding with other vehicles.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/fig4_intersection_roundabout_scenario.pdf}
\caption{Two representative driving scenarios in the road traffic junctions: (A) intersection; (B) roundabout. The ego vehicle with green colour is driving as guided by the red arrow to reach the goal.}
\label{fig_scenario_eg}
\end{figure}

\subsubsection{Intersection Scenario}

\begin{itemize}[leftmargin=*]

\item \textit{Settings}. The intersection scene comprises two roads crossing perpendicularly and stretching towards four directions (north, south, east, and west). The roads are populated with the ego vehicle and several surrounding vehicles. The positions, velocities and destinations of the other vehicles are randomly initialised. The ego vehicle drives from 60 metres south to the intersection, and the task of the ego vehicle is to cross the intersection and turn left (west). The goal will be achieved (i.e., destination arrived) if the ego vehicle could turn left at the intersection and drive 25 metres or more from the turning point in 13 seconds. Please note that the horizontal road has the right of way in the intersection scenario as for the road priority. \textbf{Figure \ref{fig_scenario_eg}-A} is a showcase of the intersection scenario where the goal of the ego vehicle with green colour is to turn left as guided by the red arrow.

\item \textit{Observations}. The vehicle observations of the intersection scenario follow the kinematic bicycle model \cite{RN755}, which implements a simplistic behaviour and considers the same lane interactions to prevent potential lateral collisions \cite{RN754}: each vehicle predicts future positions of neighbouring vehicles over a three-second horizon. If a collision with a neighbour is predicted, the yielding vehicle is decided according to the road priorities. The chosen yielding vehicle will brake until the collision prediction clears.

\item \textit{Actions}. The ego vehicle in the intersection scenario is designed to operate by selecting one from a finite set of actions \textit{A = \{slower, no-operation, faster\}}, where the vehicle can choose to slow down, keep the constant speed and accelerate.

\item \textit{Rewards}. The reward of the ego vehicle for the intersection scenario is designed as follows: the ego vehicle receives a reward of 1 if it drives at the maximum speed or if it arrives at the destination and receives a reward of -5 if a collision happens. This reward design encourages the ego vehicle to cross the intersection and arrive at the destination as soon as possible while avoiding collisions simultaneously.

\end{itemize}

\subsubsection{Roundabout Scenario}

\begin{itemize}[leftmargin=*]

\item \textit{Settings}. The scene of the roundabout scenario is composed of a two-lane roundabout with four entrances/exits (north, south, east, and west), and the roads are occupied with the ego vehicle and several surrounding vehicles. The positions, velocities and destinations of the other vehicles are initialised randomly. The ego vehicle drives from 125 metres south to the roundabout, and the task of the ego vehicle is to cross the roundabout and take the second exit (north). The goal can be accomplished if the ego vehicle can successfully take the second exit at the roundabout and drive 10 metres or more from the exit point in 13 seconds. Please note that vehicles in the roundabout have the right of way as for the road priority. As presented in \textbf{Figure \ref{fig_scenario_eg}-B} for a showcase of the roundabout scenario, the goal of the ego vehicle is to exit the roundabout following the red arrow.

\item \textit{Observations}. The vehicle observations of the roundabout scenario are applied to the kinematics type as well. The ego vehicle observes and learns from a $V*F$ array, where $V$ is the number of nearby vehicles, and $F$ is the size of features observed. In our setting, the ego vehicle observes a feature set of size 7: $\{p,x,y,v_x,v_y,cos_h,sin_h\}$, where $p$ represents whether a vehicle is in the row (whether a vehicle was observed), $x$ and $y$ describe the location of the vehicle, $v_x$ and $v_y$ denote the $x$ and $y$ axes of velocity, and $cos_h$ and $sin_h$ describe the heading direction of the vehicle.

\item \textit{Actions}. The actions of the ego vehicle in the roundabout scenario are selected from a finite set: \textit{A = \{lane\_{left}, lane\_{right}, idle, faster, slower\}}, implying that the vehicle can choose to change to the left/right lane, keep the same speed and lane, and accelerate and decelerate.

\item \textit{Rewards}. The reward of the ego vehicle for the roundabout scenario is arranged as follows: the ego vehicle receives a reward of 0.5 if it drives at the maximum speed and receives a reward of -1 if a collision happens. Every time the vehicle changes a lane, it will be awarded as -0.05. We also designed a success reward of 1 if the ego vehicle arrives at the destination, encouraging the ego vehicle to reach the destination. We suppose that this reward design encourages the vehicle to cross the roundabout and arrive at the destination as soon as possible while avoiding collisions or unnecessary lane changing simultaneously.

\end{itemize}

\subsection{Evaluation Metrics}

Our study defined four evaluation metrics heavily considering safety concerns (collision, success, and freezing) and the standard encouraging total reward to assess three baseline DRL models (DQN, PPO, and A2C) and our proposed self-awareness safety DRL: attention-DQN.

\begin{itemize}[leftmargin=*]

\item \textit{Collision rate}. It calculates the percentage of the episodes where a collision happens to the ego vehicle, reflecting the safety level of the DRL models used to train and test the performance of an autonomous vehicle.

\item \textit{Success rate}. It counts the percentage of the episodes where the ego vehicle successfully reaches the destination within the allowed duration. It also considers the functionality of the autonomous vehicle, which evaluates the ability of a DRL model to learn behaviour to complete the desired task. In our experiment, the driving scenario has a timeout setting representing the maximum duration to cross the intersection. Vehicles exceeding this duration are examined as unacceptably slow, which aligns with real-world driving scenarios where cars moving extremely slowly on the road are usually unacceptable.

\item \textit{Freezing rate}. It determines the percentage of episodes where the ego vehicle freezes or moves extremely slowly in the driving environment. In the transportation system, it is common to observe a freezing robot problem, where the ego vehicle acts overcautiously and freezes or moves extremely slowly on the road. In this case, the ego vehicle can be a serious safety hazard to traffic, such as the case of causing serial rear-end collisions. This metric is vital for straightforward performance comparisons to evaluate whether our proposed module can help alleviate the freezing robot problem, since the frozen ego vehicle will neither contribute to collision rate nor success rate. In our experiment, the freezing rate is calculated by the residual value from the success rate and collision rate as follows:

\begin{equation}
    Rate_{freezing}=100\%-Rate_{collision}-Rate_{success}
    \end{equation}

\item \textit{Total reward}. It is a typical metric commonly used to evaluate ego vehicle premium performance over several episodes. Our experiment collected the total reward that the ego vehicle receives after taking a set of actions in each episode. As the ego vehicle learns to drive better, the total reward the ego vehicle acquires in each episode would increase with more learning episodes.

\end{itemize}

\subsection{Experimental Procedure}

We conducted two experiments in this study: in experiment 1, three baseline DRL models, DQN, A2C, and PPO, were selected to train the ego vehicle and then evaluate the safety performance based on four defined metrics in the intersection and roundabout driving scenarios; in experiment 2, we trained the ego vehicle in the intersection and roundabout driving scenarios using the proposed single-query and multi-query attention-DQN to compare the performance with the baseline DQN model. In addition, we presented the safety performance of the above DRL models from two experiments during the training and testing phases.

\subsubsection{Experiment 1}

During the training phase, the ego vehicle was drilled by three baseline DRL models (DQN, A2C, and PPO) in intersection and roundabout driving scenarios. We observed that the collision rate and success rate in each DRL model converged after 60,000 training episodes with 5 runs. In the testing phase, we evaluated the trained DRL models over 100 episodes for 100 trials to collect the testing data and tested each for 20 trials. For each trial, the number of collisions and successes during the 100 testing episodes were recorded to represent the collision rate and success rate. Additionally, the freezing rate was calculated through the residual value from the collected collision rate and success rate.

\subsubsection{Experiment 2}

We trained the ego vehicle using default single-query and multi-query attention-DQN in intersection and roundabout driving scenarios for 60,000 episodes with 5 runs. Please note that the hyperparameters we used for attention-based DQN are the same as the baseline DQN for a fair comparison. In the testing phase, we evaluated the trained models from single-query and multi-query attention-DQN over 100 episodes for 100 trials to collect the testing data and tested each for 20 trials. We recorded the number of collisions and the number of successes during the 100 episodes to indicate the collision rate and success rate for each trial. The freezing rate is the calculated residual value from the collision rate and the success rate.
\section{Results}

\subsection{Evaluation of Experiment 1}

\subsubsection{Training Phase}

\begin{itemize}[leftmargin=*]

\item \textit{Intersection scenario}. As shown in \textbf{Figure \ref{fig_exp1_intersection}}, we presented the training evolution performance of collision rate (\textbf{Figure \ref{fig_exp1_intersection}-A}), success rate (\textbf{Figure \ref{fig_exp1_intersection}-B}), freezing rate (\textbf{Figure \ref{fig_exp1_intersection}-C}) and total reward (\textbf{Figure \ref{fig_exp1_intersection}-D}) of the ego vehicle in the intersection scenario, trained by three baseline DRL models (DQN, PPO, and A2C) over 60,000 episodes and repeated 5 times, and the displayed values were averaged with 95\% confidence interval. We found that the collision rate of the ego vehicles trained by three baseline DRL models reached above 60\% at the end of the training steps, particularly in DQN, which revealed the lowest collision rate during the training process. We also noticed that the success rates of PPO and A2C reached approximately the same value at 35\%, and the success rate of DQN reached approximately 28\%, which was reflected in the total reward curves. In addition, the freezing rates of the ego vehicles trained by PPO and A2C reached approximately 0, starting from 15,000 episodes, whereas the freezing rate of DQN decreased to approximately 8\%, being the best performance among all baseline DRL models.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/exp1_training_inter.pdf}
\caption{Intersection scenario: the training performance of the ego vechile with four metrics: (A) collision rate, (B) success rate, (C) freezing rate, and (D) total reward using the baseline DRL models (DQN, A2C, and PPO).}
\label{fig_exp1_intersection}
\end{figure}

\item \textit{Roundabout scenario}. As presented in \textbf{Figure \ref{fig_exp1_roundabout}}, we demonstrated the training evolution of the performance of the collision rate (\textbf{Figure \ref{fig_exp1_roundabout}-A}), success rate (\textbf{Figure \ref{fig_exp1_roundabout}-B}), freezing rate (\textbf{Figure \ref{fig_exp1_roundabout}-C}) and total reward (\textbf{Figure \ref{fig_exp1_roundabout}-D}) of the ego vehicle during training in the roundabout scenario, trained by three baseline DRL models (DQN, PPO, and A2C) over 60,000 episodes and repeated 5 times, and the displayed values were averaged with a 95\% confidence interval. We recognised that the collision rate of the ego vehicles trained by all baseline DRL models was below 25\% at the end of the training, in particular A2C being visible as the lowest at approximately 5\%. In contrast, the freezing rates of the ego vehicles trained by A2C and DQN both reached above 60\%, specific to A2C being approximately 73\% and DQN being approximately 63\%. In addition, PPO revealed the lowest freezing rate and the highest success rate in baseline DRL models.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/exp1_training_round.pdf}
\caption{Roundabout scenario: the training performance of the ego vechile with four metrics: (A) collision rate, (B) success rate, (C) freezing rate, and (D) total reward using the baseline DRL models (DQN, A2C, and PPO).}
\label{fig_exp1_roundabout}
\end{figure}

\end{itemize}

\subsubsection{Testing Phase}

\textbf{Table \ref{table_1}} shows the collision rates, success rates, and freezing rates of the ego vehicles tested in the intersection and roundabout scenario using three baseline DRL models over 100 episodes and averaged across 100 runs (10,000 testing episodes in total for each model). Regarding the intersection scenario, the collision rates of the ego vehicles trained by all three baseline models were above 50\%, with PPO being the highest at 64.45\% and DQN being the lowest at 56.88\%. In terms of success rate, DQN also came the lowest value, at 29.23\%. This was caused by DQN being the only case that experienced a freezing robot problem, with a freezing rate of 13.89\%. The PPO and A2C models did not have freezing vehicles during the testing phase. Moving to the testing phase of the roundabout scenario, A2C demonstrated the lowest collision rate of 7.2\%, followed by DQN of 14.7\% and PPO of 20.47\%. DQN had the lowest success rate and the highest freezing rate, whereas PPO had the highest success rate and lowest freezing rate, with A2C being the second in both metrics.

\begin{table}[!ht]
\caption {The testing performance of three baseline DRL models in intersection and roundabout scenarios.} \label{table_1}
\begin{adjustbox}{width=\columnwidth,center}
\begin{tabular}{@{}ccccc@{}}
\toprule
\textbf{Scenarios}                                           & \textbf{\begin{tabular}[c]{@{}c@{}}Metrics\\ mean (std) \%\end{tabular}} & \textbf{DQN}  & \textbf{A2C} & \textbf{PPO} \\ \midrule
\multicolumn{1}{c|}{\multirow{3}{*}{\textbf{Intersection}}} & \textit{Collision Rate}                                             & 56.88 (5.39)  & 65.51 (5.08) & 64.45 (4.93) \\
\multicolumn{1}{c|}{}                                       & \textit{Success Rate}                                               & 29.23 (4.99)  & 34.49 (5.08) & 35.54 (4.93) \\
\multicolumn{1}{c|}{}                                       & \textit{Freezing Rate}                                              & 13.89 (3.81)  & 0 (0)        & 0 (0)        \\ \midrule
\multicolumn{1}{c|}{\multirow{3}{*}{\textbf{Roundabout}}}   & \textit{Collision Rate}                                             & 14.7 (7.24)   & 7.2 (3.03)   & 20.47 (4.78) \\
\multicolumn{1}{c|}{}                                       & \textit{Success Rate}                                               & 33.99 (5.66)  & 46.35 (7.70) & 74.42 (4.91) \\
\multicolumn{1}{c|}{}                                       & \textit{Freezing Rate}                                              & 51.31 (10.33) & 46.45 (8.58) & 5.11 (5.39)  \\ \cmidrule(l){2-5}
\end{tabular}
\end{adjustbox}
\end{table}

\subsubsection{Our Findings}

We observed significant room to improve safety performance in the testing phase, as shown in \textbf{Table \ref{table_1}}, especially the collision rate of the DRL models in DQN. In the intersection scenario, we observed a high collision rate - more than 50\% from all three baseline DRL models (56.88\% for DQN, 65.51\% for A2C, and 64.45\% for PPO), suggesting that the autonomous vehicle in the road traffic junction driving environment is far from safe. Furthermore, we noticed that the collision rate in the roundabout scenario is lower (<20\%) than that in the intersection scenario, but it is mainly caused by the high freezing rate that does not indeed contribute to safety concerns. Specifically, the freezing robot problem observed in the roundabout scenario was remarked during the training phase of all baseline DRL models, especially to DQN, which posed more than half of the freezing episodes (51.31\%). Considering the above performance results, we chose DQN to investigate further whether the proposed self-attention module could decrease the collision rate and alleviate the freezing robot problem in intersection and roundabout driving scenarios.

In addition, we demonstrated a showcase in \textbf{Figure \ref{fig_exp1_findings}} about testing episodes using DQN in intersection and roundabout scenarios to further clarify our findings. The goal of the ego vehicle is to turn left in the intersection scenario (\textbf{Figure \ref{fig_exp1_findings}-A}), and the training behaviours of the ego vehicle follows: (1) the ego vehicle began from the starting point, driving towards the intersection; (2) the ego vehicle reached the intersection entry, and some other vehicles came from left and right. The surrounding vehicle on the left probably would not collide with the ego vehicle since it was turning right, but the surrounding vehicle from the right might collide with the ego vehicle; (3) the ego vehicle made the decision to turn left and collided with the incoming car from the right. For the roundabout scenario (\textbf{Figure \ref{fig_exp1_findings}-B}), the goal of the ego vehicle is to take the north (second) exit from the roundabout, and the training behaviours of the ego vehicle follow as follows: (1) the ego vehicle began from the starting point, driving towards the roundabout, and there were other vehicles in the roundabout that might collide with the ego vehicle; (2) the ego vehicle stayed basically where it was when it seemed safe to enter the roundabout; (3) the roundabout was completely clear of vehicles, and the ego vehicle finally moved forward before time was out, and the episode ended.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/exp1_findings.pdf}
\caption{A showcase of testing episodes using DQN in (A) intersection and (B) roundabout scenarios.}
\label{fig_exp1_findings}
\end{figure}

\subsection{Evaluation of Experiment 2}

\subsubsection{Training Phase}

\begin{itemize}[leftmargin=*]

\item \textit{Intersection scenario}. As shown in \textbf{Figure \ref{fig_exp2_intersection}}, we presented the training evolution of collision rate (\textbf{Figure \ref{fig_exp2_intersection}-A}), success rate (\textbf{Figure \ref{fig_exp2_intersection}-B}), freezing rate (\textbf{Figure \ref{fig_exp2_intersection}-C}) and total reward (\textbf{Figure \ref{fig_exp2_intersection}-D}) of the ego vehicle in the intersection scenario, learned by DQN, default (single-query) and multiQ (multi-query) attention-DQN over 60,000 episodes, which displayed by the mean values averaged from 5 runs, with 95\% confidence interval. We observed that the ego vehicle trained by the default attention-DQN (the orange colour) achieved the lowest collision rate and the highest success rate, which was demonstrated by a significant drop in the collision rate, from approximately 67\% of the DQN to approximately 20\% of the attention-DQN, as well as by a considerable increase from 28\% of DQN to 49\% of the attention-DQN. The changes in collision and success rates were also reflected in the total reward. We also noticed that the multiQ-attention-DQN (the green colour) exhibited a similar performance yet not as strong compared with default attention-DQN.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/exp2_training_inter.pdf}
\caption{Intersection scenario: the training performance of the ego vechile with four metrics: (A) collision rate, (B) success rate, (C) freezing rate, and (D) total reward using DQN, single-query and multi-query attention-DQN.}
\label{fig_exp2_intersection}
\end{figure}

\item \textit{Roundabout scenario}. \textbf{Figure \ref{fig_exp2_roundabout}} demonstrates the training evolution of the collision rate (\textbf{Figure \ref{fig_exp2_roundabout}-A}), success rate (\textbf{Figure \ref{fig_exp2_roundabout}-B}), freezing rate (\textbf{Figure \ref{fig_exp2_roundabout}-C}) and total reward (\textbf{Figure \ref{fig_exp2_roundabout}-D}) of the ego vehicles in the roundabout scenario learned by DQN, default (single-query) and multiQ (multi-query) attention-DQN over 60,000 episodes, which are displayed by the mean values averaged from 5 runs, with a 95\% confidence interval. We observed that the ego vehicle trained by attention-DQN was quickest to avoid collisions during the beginning training period, such as the curve's steepness between 1 to 10,000 episodes, and then reached the low collision rate of 12\%at episode 60,000 finally. For the success rate, the ego vehicle trained by the attention-DQN achieved the highest performance by presenting approximately 74\%, which is a significant improvement compared to 27\% introduced by DQN. In addition, the freezing rate dropped from 63\% of DQN to approximately 15\% of attention-DQN.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/exp2_training_round.pdf}
\caption{Roundabout scenario: the training performance of the ego vechile with four metrics: (A) collision rate, (B) success rate, (C) freezing rate, and (D) total reward using DQN, single-query and multi-query attention-DQN.}
\label{fig_exp2_roundabout}
\end{figure}

\end{itemize}

\subsubsection{Testing Phase}

As shown in \textbf{Table \ref{table_2}}, we summarised the collision rate, success rate, and freezing rate of the ego vehicle tested by trained DRL models (DQN, default attention-DQN and MultiQ-attention-DQN) over 100 episodes and averaged across 100 runs (10,000 testing episodes in total for each DRL model) in the intersection and roundabout scenarios. In the intersection scenario, the testing performance of the ego vehicle trained by attention-DQN showed the lowest collision rate of 14.57\% and the highest success rate of 52.58\%. For the roundabout scenario, similarly, attention-DQN achieved the lowest collision rate of 12.80\% and presented a significant increase in the success rate, from 33.99\% of DQN to 83.92\% of attention-DQN. The freezing rate of the attention-DQN unveiled the lowest value of 3.28\%, which was reduced significantly compared to that of DQN (51.31\%). We also noticed that the performance of multiQ-attention-DQN ranked second, slightly worse than attention-DQN, but very close.

\begin{table}[!ht]
\caption {The testing performance of DQN, single-query and multi-query attention-DQN in intersection and roundabout scenarios.} \label{table_2}
\begin{adjustbox}{width=\columnwidth,center}
\begin{tabular}{@{}c|cccc@{}}
\toprule
\multirow{2}{*}{\textbf{Scenarios}}    & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Metrics\\ mean (std) \%\end{tabular}}} & \multirow{2}{*}{\textbf{DQN}} & \multicolumn{2}{c}{\textbf{Attention-DQN}}   \\ \cmidrule(l){4-5}
                                       &                                         &                               & \textbf{Default} & \textbf{MultiQ} \\ \midrule
\multirow{3}{*}{\textbf{Intersection}} & \textit{Collision Rate}               & 56.88 (5.39)   & 14.57 (3.76)  & 18.54 (6.40)                 \\
                                       & \textit{Success Rate}                 & 29.23 (4.99)   & 52.58 (6.35)  & 49.18 (6.74)                 \\
                                       & \textit{Freezing Rate}                & 13.89 (3.81)   & 32.85 (6.03)  & 32.28 (8.88)                 \\ \midrule
\multirow{3}{*}{\textbf{Roundabout}}   & \textit{Collision Rate}               & 14.70 (7.24)   & 12.80 (5.30)  & 12.93 (4.73)                 \\
                                       & \textit{Success Rate}                 & 33.99 (5.66)   & 83.92 (4.30)  & 80.90 (8.06)                 \\
                                       & \textit{Freezing Rate}                & 51.31 (10.33)  & 3.28 (3.17)   & 6.17 (9.93)
\end{tabular}
\end{adjustbox}
\end{table}

\subsubsection{Our Findings}

The evaluation outcomes of attention-DQN effectively found improved safety concerns of the autonomous vehicle in intersection and roundabout driving scenarios. In the intersection scenario, translation as the attention mechanism helped the ego vehicle pay better attention to the other vehicles in the traffic, thus avoiding collisions and completing the task more efficaciously and successfully. In the roundabout scenario, attention-DQN showed a meagre freezing rate of ego vehicles of only 3.28\% in the test phase, suggesting that the attention mechanism alleviates the problem of robot freezing because the ego vehicle is more confident in moving forward with supplementary attention information to the surrounding vehicles.

\begin{figure}[!htp]
\centering
\includegraphics [width=\columnwidth] {figures/exp2_findings.pdf}
\caption{A showcase of testing episodes using attention-DQN in (A) intersection and (B) roundabout scenarios. The two attention heads are visualised as green and blue lines connecting the ego vehicle and other surrounding vehicles, and the width of a line corresponds to the weight of the attention. }
\label{fig_exp2_findings}
\end{figure}

In addition, we demonstrated a showcase
\footnote {A testing demonstration of replay videos is available at \href{https://github.com/czh513/DRL_Self-Awareness-Safety}{GitHub}. }
in \textbf{Figure \ref{fig_exp2_findings}} about testing episodes using attention-DQN in intersection and roundabout scenarios to further clarify our findings. Here, please note that "attention" is visualised as lines connecting the ego vehicle and other surrounding vehicles. The two different colours (green and blue) represent two attention heads, and the width of the line corresponds to the weight of the attention. The goal of the ego vehicle is to turn left in the intersection scenario (\textbf{Figure \ref{fig_exp2_findings}-A}), and the training attention behaviours of the ego vehicle follow as: (1) the ego vehicle paid attention to the surrounding vehicles from left, front, and right directions; (2) when ego vehicle coming closer to the intersection, the attention to the incoming other vehicles strengthened; (3) the surrounding vehicles from the left to right turn or from the front to left turn at the intersection that were no longer threats to the ego vehicle, so the main attention switched to the surrounding vehicles from the right whose intention was not clear yet;  (4) becasue the vehicles from the right showed intention to turn right and drove almost out of the intersection, the attention on it decreased; (5) the intersection was clear for all surrounding vehicles, so that the attention came back to the ego vehicle itself when the ego vehicle crossed the intersection; (6) the ego vehicle drove on the destination lane with slight attention to the vehicle in front of it to keep distance. Moreover, targetting to the roundabout scenario (\textbf{Figure \ref{fig_exp2_findings}-B}), the ego vehicle needs to learn to cross the roundabout and take the desired exit, and the training attention behaviours of the ego vehicle follow as follows: (1) the ego vehicle paid substantial attention to the surrounding vehicles coming from the left that were likely to collide with the ego vehicle; (2) the ego vehicle waited until the surrounding vehicle from left passed the entry point and switched its primary attention to the surrounding vehicles coming from the following entry; (3) the ego vehicle entered the roundabout and kept its attention on other surrounding vehicles; (4) the attention to the front vehicle strengthened, since the ego vehicle was getting closer to its front vehicle; (5) the ego vehicle exited the roundabout but still kept its attention on the front vehicle to keep the reasonable distance; and (6) the attention switched back to the ego vehicle when it was safe enough.

\section{Discussion}

Based on our findings, we observed that Attention-DQN lowered the freezing rate of the ego vehicle significantly in the roundabout scenario, from above 50\% freezing rate in both training and testing results to only 3.28\% in testing and 15\% in training. This translates as a significant improvement on the freezing robot problems in the roundabout scenario. One possible reason we speculated that Attention-DQN helped to achieve a low freezing rate is that the attention mechanism enables the ego vehicle to be more confident in moving forward by paying more attention to the surrounding vehicles. Interestingly, the attention mechanism did not have the same effect on the ego vehicle in the intersection scenario. The freezing rates of both training and testing results after the applied attention mechanism were higher than the baseline algorithm. The collision rate did decrease a lot in the intersection scenario, but not all of the decreased percentage effectively transferred to the success rate and some part of the collision rate transferred to the freezing rate. It is unclear in the current stage, but one observation during the experiments was that crashes between other vehicles happened in intersections which sometimes blocked the path of the ego vehicle. In this case, the ego vehicle without attention collided with the crashed vehicles, but with attention, it has had to choose to freeze, as referred in \textbf{Figure \ref{fig_11}}.

\begin{figure}[!htp]
\centering
\includegraphics [width=4cm] {figures/fig11.pdf}
\caption{A showcase of the frozen vehicle in intersection scenario.}
\label{fig_11}
\end{figure}

For DRL in a complex autonomous driving environment, our research still presented two limitations: The simulated driving environment might exhibit stochastic behaviours that included our testing driving scenarios are complex scenarios populated with vehicles where one vehicle on the road may collide with another. During the experiments, we observed that sometimes there were crashes between other vehicles, which blocked the path of our ego vehicle. As a result, the ego vehicle could not move forward and could only wait till timeout, but it was used to calculate into freezing rate, suggesting that the freezing rate might be influenced by this case. Also, our proposed model is still not entirely safe to be applied to real-world autonomous driving, so it is worth exploring to combine optimisation measurements and more sensing information with DRL trained agents for further safety improvement.

\section{Conclusion}

In this study, we evaluated the safety performance of autonomous driving in a complex road traffic junction environment, such as challenging intersection and roundabout scenarios. We designed four evaluation metrics, including collision rate, success rate, freezing rate, and total reward, from three baseline DRL models (DQN, A2C, and PPO) to the proposed attention-DQN module for self-awareness safety improvement. Our results based on two experiments showed drawbacks of the current DRL models and presented significantly reduced collision rate and freezing rate by introducing the attention mechanism that validated the effectiveness of our proposed attention-DQN module in enhancing the safety concerns of the ego vehicle. Our findings have the potential to contribute to and benefit safe DRL in transportation applications.

\balance
\bibliographystyle{IEEEtran}


\section{Introduction}
\label{intro}

The anomalous magnetic moment of the muon $a_\mu$ provides an
important test of the Standard Model (SM) and is potentially sensitive
to contributions from New Physics~\cite{JN_09}. In fact, for several
years now a deviation is observed between the experimental measurement
and the SM prediction, $a_\mu^{\rm exp} - a_\mu^{\rm SM} \sim
(250-300) \times 10^{-11}$, corresponding to about $3 - 3.5$~standard
deviations~\cite{JN_09,recent_estimates}.

Hadronic effects dominate the uncertainty in the SM prediction of
$a_\mu$ and make it difficult to interpret this discrepancy as a sign
of New Physics. In particular, in contrast to the hadronic vacuum
polarization in the $g-2$, which can be related to data, the estimates
for the hadronic light-by-light (LbyL) scattering contribution
$a_\mu^{{\rm had.\ LbyL}} = (105 \pm 26) \times
10^{-11}$~\cite{PdeRV_09} and $a_\mu^{{\rm had.\ LbyL}} = (116 \pm 40)
\times 10^{-11}$~\cite{Nyffeler_09,JN_09} rely entirely on
calculations using {\it hadronic models} which employ form factors for
the interaction of hadrons with photons. The more recent
papers~\cite{Schwinger_Dyson,GdeR_12} yield a larger central value and
a larger error of about $(150 \pm 50) \times 10^{-11}$. For a brief
review of had.\ LbyL scattering in $a_\mu$ see Ref.~\cite{B_ZA}, which
also includes a reanalysis of the charged pion loop contribution, see
also Ref.~\cite{pion_loop}.  To fully profit from future planned $g-2$
experiments with a precision of $15 \times 10^{-11}$, these large
model uncertainties have to be reduced. Maybe lattice QCD will at some
point give a reliable number, see the talk~\cite{Blum_Lattice_2012}
for some encouraging progress recently. Meanwhile, experimental
measurements and theoretical constraints of the relevant form factors
can help to constrain the models and to reduce the uncertainties in
$a_\mu^{{\rm had.\ LbyL}}$.

In most model calculations, pion-exchange gives the numerically
dominant contribution\footnote{Apart from the recent
  papers~\cite{Schwinger_Dyson,GdeR_12}, where the (dressed) quark
  loop gives the largest contribution.}, therefore it has received a
lot of attention. In our paper~\cite{KLOE-2_impact} we studied the
impact of planned measurements at KLOE-2 of the $\pi^0\to\gamma\gamma$
decay width to 1\% statistical precision and the
$\gamma^\ast\gamma\to\pi^0$ transition form factor ${\cal
  F}_{\pi^0\gamma^\ast\gamma}(Q^2)$ for small space-like momenta,
$0.01~\mbox{GeV}^2 \leq Q^2 \leq 0.1~\mbox{GeV}^2$, to 6\% statistical
precision in each bin, on estimates of the pion-exchange contribution
$a_\mu^{{\rm LbyL}; \pi^0}$.  We would like to stress that a realistic
calculation of $a_\mu^{{\rm LbyL}; \pi^0}$ is {\it not} the purpose of
this paper.  The estimates given below are performed to demonstrate,
within several models, an improvement of uncertainty, which will be
possible when the KLOE-2 data appear. The simulations in
Ref.~\cite{KLOE-2_impact} have been performed with the dedicated
Monte-Carlo program EKHARA~\cite{EKHARA} for the process $e^+ e^- \to
e^+ e^- \gamma^* \gamma^* \to e^+ e^- P$ with $P = \pi^0, \eta,
\eta^\prime$, followed by the decay $\pi^0 \to \gamma\gamma$ and
combined with a detailed detector simulation.

\section{Impact of KLOE-2 measurements on $a_\mu^{\mathrm{LbyL};\pi^0}$}
\label{sec:KLOE}

Any experimental information on the neutral pion lifetime and the
transition form factor is important in order to constrain the models
used for calculating the pion-exchange contribution.  However, having
a good description e.g.\ for the transition form factor is only
necessary, not sufficient, in order to uniquely determine
$a_\mu^{\mathrm{LbyL};\pi^0}$. As stressed in
Ref.~\cite{Jegerlehner_off-shell}, what enters in the calculation of
$a_\mu^{{\rm LbyL}; \pi^0}$ is the fully off-shell form factor ${\cal
  F}_{{\pi^0}^*\gamma^*\gamma^*}((q_1 + q_2)^2, q_1^2, q_2^2)$ (vertex
function), where also the pion is off-shell with 4-momentum $(q_1 +
q_2)$. Such a (model dependent) form factor can for instance be
defined via the QCD Green's function $\langle VVP \rangle$, see
Ref.~\cite{Nyffeler_09} for details and references to earlier work.
The form factor with on-shell pions is then given by ${\cal
  F}_{\pi^0\gamma^*\gamma^*}(q_1^2, q_2^2) \equiv {\cal
  F}_{{\pi^0}^*\gamma^*\gamma^*}(m_\pi^2, q_1^2, q_2^2)$.
Measurements of the transition form factor ${\cal
  F}_{\pi^0\gamma^\ast\gamma}(Q^2) \equiv {\cal
  F}_{{\pi^0}^\ast\gamma^\ast\gamma^\ast}(m_{\pi}^2, -Q^2, 0)$ are in
general only sensitive to a subset of the model parameters and do not
allow to reconstruct the full off-shell form factor.

For different models, the effects of the off-shell pion can vary a
lot. In Ref.~\cite{Nyffeler_09} an off-shell form factor (LMD+V) was
proposed, based on large-$N_C$ QCD matched to short-distance
constraints from the operator product expansion, see also
Ref.~\cite{KN_EPJC_2001}. This yields the estimate $a_{\mu; {\rm
    LMD+V}}^{{\rm LbyL}; \pi^0} = (72 \pm 12) \times 10^{-11}$. The
error estimate comes from the variation of all model parameters, where
the uncertainty of the parameters related to the off-shellness of the
pion completely dominates the total error and will {\it not} be shown
in Table~\ref{tab:amu} below.

In contrast to the off-shell LMD+V model, many models, e.g.\ the VMD
model, constituent quark models or the ans\"atze for the transition
form factor used in Ref.~\cite{Cappiello:2010uy}, do not have these
additional sources of uncertainty related to the off-shellness of the
pion. These models often have only very few parameters, which can all
be fixed by measurements of the transition form factor or from other
observables. Therefore, the precision of the KLOE-2 measurement can
dominate the total accuracy of $a_\mu^{\mathrm{LbyL};\pi^0}$ in such
models.

It was noted in Ref.~\cite{Nyffeler:2009uw} that essentially all
evaluations of the pion-exchange contribution use the normalization
${\cal F}_{{\pi^0}^*\gamma^*\gamma^*}(m_\pi^2, 0, 0) = 1 / (4 \pi^2
F_\pi)$ for the form factor, as derived from the Wess-Zumino-Witten
(WZW) term. Then the value $F_\pi = 92.4~\mbox{MeV}$ is used without any
error attached to it, i.e. a value close to $F_\pi = (92.2 \pm
0.14)~\mbox{MeV}$, obtained from $\pi^+ \to \mu^+
\nu_\mu(\gamma)$~\cite{Nakamura:2010zzi}. Instead, if one uses the
decay width $\Gamma_{\pi^0 \to \gamma\gamma}$ for the normalization of
the form factor, an additional source of uncertainty enters, which has
not been taken into account in most evaluations.

In our calculations we account for this normalization issue, using in
the fit:
\begin{itemize}
\item $\Gamma^{{\rm PDG}}_{\pi^0 \to \gamma\gamma} = 7.74 \pm
0.48$~eV from the PDG 2010~\cite{Nakamura:2010zzi},
\item $\Gamma^{{\rm PrimEx}}_{\pi^0 \to \gamma\gamma} = 7.82 \pm
0.22$~eV from the PrimEx experiment~\cite{Larin:2010kq},
\item $\Gamma^{{\rm KLOE-2}}_{\pi^0 \to \gamma\gamma} = 7.73 \pm
0.08$~eV for the KLOE-2 simulation (assuming a $1\%$ precision).
\end{itemize}

The assumption that the KLOE-2 measurement will be consistent with the
LMD+V and VMD models, allowed us in Ref.~\cite{KLOE-2_impact} to use
the simulations as new ``data'' and evaluate the impact of such
``data'' on the precision of the $a_\mu^{{\rm LbyL}; \pi^0}$
calculation.  In order to do that, we fit the LMD+V and VMD models to
the data sets~\cite{TFF_data} from CELLO, CLEO and BaBar for the
transition form factor and the values for the decay width given above:
\begin{displaymath}
\label{eq:fitdatasets}
 \begin{array}{ll}
   A0: & \mbox{CELLO, CLEO, PDG} \\
   A1: & \mbox{CELLO, CLEO, PrimEx} \\
   A2: & \mbox{CELLO, CLEO, PrimEx, KLOE-2} \\
   B0: & \mbox{CELLO, CLEO, BaBar, PDG} \\
   B1: & \mbox{CELLO, CLEO, BaBar, PrimEx} \\
   B2: & \mbox{CELLO, CLEO, BaBar, PrimEx, KLOE-2}
 \end{array}
\end{displaymath}

The BaBar measurement of the transition form factor does not show the
$1/Q^2$ behavior as expected from earlier theoretical considerations
by Brodsky-Lepage~\cite{Brodsky-Lepage} and as seen in the CELLO and
CLEO data and the recent measurements from Belle~\cite{Belle}.  The
VMD model always shows a $1/Q^2$ fall-off and therefore is not
compatible with the BaBar data. The LMD+V model has another parameter,
$h_1$, which determines the behavior of the transition form factor for
large $Q^2$. To get the $1/Q^2$ behavior according to Brodsky-Lepage,
one needs to set $h_1 = 0$. However, one can simply leave $h_1$ as a
free parameter and fit it to the BaBar data, yielding $h_1 \neq
0$~\cite{Nyffeler:2009uw}. In this case the form factor does not
vanish for $Q^2 \to \infty$. Since VMD and LMD+V with $h_1 = 0$ are
not compatible with the BaBar data, the corresponding fits are very
bad and we will not include these results in the current paper, see
Ref.~\cite{KLOE-2_impact} for details.

For illustration, we use the following two approaches to calculate
 $a_\mu^{{\rm LbyL}; \pi^0}$:
\begin{itemize}
 \item Jegerlehner-Nyffeler (JN)
       approach~\cite{Nyffeler_09,JN_09}
       with the off-shell pion form factor;
 \item Melnikov-Vainshtein (MV) approach~\cite{Melnikov:2003xd},
       where one uses the on-shell pion form factor at the internal
       vertex and a constant (WZW) form factor at the external vertex.
\end{itemize}

Table~\ref{tab:amu} shows the impact of the PrimEx and the future
KLOE-2 measurements on the model parameters and, consequently, on the
$a_\mu^{{\rm LbyL}; \pi^0}$ uncertainty. The other parameters of the
(on-shell and off-shell) LMD+V model have been chosen as in the
papers~\cite{Nyffeler_09,JN_09,Melnikov:2003xd}.  We stress again that
our estimate of the $a_\mu^{{\rm LbyL}; \pi^0}$ uncertainty is given
only by the propagation of the errors of the fitted parameters in
Table~\ref{tab:amu} and therefore we may not reproduce the total
uncertainties given in the original papers.

\begin{table*}
 \caption{KLOE-2 impact on the accuracy of $a_\mu^{{\rm
       LbyL}; \pi^0}$ in case of one year of data taking
   ($5$~fb$^{-1}$). The values marked with asterisk (*)
   do not contain additional uncertainties coming from
   the ``off-shellness'' of the pion (see the text).}
 \label{tab:amu}
 {\scriptsize
 \begin{tabularx}{\textwidth}{clcllll}
 Model&Data& $\chi^2/d.o.f.$ &  & Parameters && $a_\mu^{{\rm LbyL};
   \pi^0} \times 10^{11}$\\
 \hline
 VMD  & A0 & $6.6/19$
     & $M_V = 0.778(18)$~GeV & $F_\pi = 0.0924(28)$~GeV  && $(57.2 \pm
 4.0)_{JN}$\\
 VMD  & A1 & $6.6/19$
     & $M_V = 0.776(13)$~GeV & $F_\pi = 0.0919(13)$~GeV  && $(57.7 \pm
 2.1)_{JN}$\\
 VMD  & A2 & $7.5/27$
     & $M_V = 0.778(11)$~GeV & $F_\pi = 0.0923(4)$~GeV   && $(57.3 \pm
 1.1)_{JN}$\\
 \hline
 LMD+V, $h_1 = 0$  & A0 & $6.5/19$
     &  $\bar{h}_5 = 6.99(32)$~GeV$^4$ & $\bar{h}_7 =
 -14.81(45)$~GeV$^6$ && $(72.3 \pm 3.5)_{JN}^*$\\
 &  &   &                        &                            &&
 $(79.8 \pm 4.2)_{MV}$\\
 LMD+V, $h_1 = 0$  & A1 & $6.6/19$
     &  $\bar{h}_5 = 6.96(29)$~GeV$^4$ & $\bar{h}_7 =
 -14.90(21)$~GeV$^6$ && $(73.0 \pm 1.7)_{JN}^*$\\
 &  &   &                        &                            &&
 $(80.5 \pm 2.0)_{MV}$\\
 LMD+V, $h_1 = 0$  & A2 & $7.5/27$
     &  $\bar{h}_5 = 6.99(28)$~GeV$^4$ & $\bar{h}_7 =
 -14.83(7)$~GeV$^6$ && $(72.5 \pm 0.8)_{JN}^*$\\
 &  &   &                        &                           && $(80.0
 \pm 0.8)_{MV}$\\
 \hline
 LMD+V, $h_1 \neq 0$  & A0 & $6.5/18$
     &  $\bar{h}_5 = 6.90(71)$~GeV$^4$ & $\bar{h}_7 = -14.83(46)$~GeV$^6$&
        $h_1 = -0.03(18)$~GeV$^2$ & $(72.4 \pm 3.8)_{JN}^*$\\
 LMD+V, $h_1 \neq 0$  & A1 & $6.5/18$
     &  $\bar{h}_5 = 6.85(67)$~GeV$^4$ & $\bar{h}_7 = -14.91(21)$~GeV$^6$&
        $h_1 = -0.03(17)$~GeV$^2$ & $(72.9 \pm 2.1)_{JN}^*$\\
 LMD+V, $h_1 \neq 0$  & A2 & $7.5/26$
     &  $\bar{h}_5 = 6.90(64)$~GeV$^4$ & $\bar{h}_7 = -14.84(7)$~GeV$^6$ &
        $h_1 = -0.02(17)$~GeV$^2$ & $(72.4 \pm 1.5)_{JN}^*$\\
 LMD+V, $h_1 \neq 0$  & B0 & $18/35$
     &  $\bar{h}_5 = 6.46(24)$~GeV$^4$ & $\bar{h}_7 = -14.86(44)$~GeV$^6$ &
        $h_1 = -0.17(2)$~GeV$^2$ & $(71.9 \pm 3.4)_{JN}^*$\\
 LMD+V, $h_1 \neq 0$  & B1 & $18/35$
     &  $\bar{h}_5 = 6.44(22)$~GeV$^4$ & $\bar{h}_7 = -14.92(21)$~GeV$^6$ &
        $h_1 = -0.17(2)$~GeV$^2$ & $(72.4 \pm 1.6)_{JN}^*$\\
 LMD+V, $h_1 \neq 0$  & B2 & $19/43$
     &  $\bar{h}_5 = 6.47(21)$~GeV$^4$ & $\bar{h}_7 = -14.84(7)$~GeV$^6$ &
        $h_1 = -0.17(2)$~GeV$^2$ & $(71.8 \pm 0.7)_{JN}^*$ \\
\hline
 \end{tabularx}
}
\end{table*}

We can clearly see from Table~\ref{tab:amu} that for each given model
and each approach (JN or MV), there is a trend of reduction in the
error for $a_\mu^{{\rm LbyL}; \pi^0}$ (related only to the given model
parameters) by about half when going from A0 (PDG) to A1 (including
PrimEx) and by about another half when going from A1 to A2 (including
KLOE-2). Very roughly, we can write:
\begin{itemize}
\item
Sets A0, B0: $\delta a_\mu^{{\rm LbyL}; \pi^0} \approx 4 \times
10^{-11}$ (with $\Gamma^{{\rm PDG}}_{\pi^0 \to \gamma\gamma}$)
\item
Sets A1, B1: $\delta a_\mu^{{\rm LbyL}; \pi^0} \approx2 \times
10^{-11}$
(with $\Gamma^{{\rm PrimEx}}_{\pi^0 \to \gamma\gamma}$)
\item
Sets A2, B2: $\delta a_\mu^{{\rm LbyL}; \pi^0} \approx (0.7 - 1.1)
\times 10^{-11}$ (with simulated KLOE-2 data)
\end{itemize}

This is mainly due to the improvement in the normalization of the form
factor, related to the decay width $\pi^0 \to \gamma\gamma$,
controlled by the parameters $F_\pi$ or $\bar{h}_7$, respectively, but
more data also better constrain the other model parameters $M_V$ or
$\bar{h}_5$. This trend of improvement is also visible in the last
part of the Table (LMD+V, $h_1 \neq 0$), when we fit the sets B0, B1
and B2 which include the BaBar data.  The central values of the final
results for $a_\mu^{{\rm LbyL}; \pi^0}$ are only slightly changed, if
we include the BaBar data.  They shift only by about $-0.5 \times
10^{-11}$ compared to the corresponding data sets A0, A1 and A2.  This
is due to a partial compensation in $a_\mu^{{\rm LbyL}; \pi^0}$, when
the central values for $\bar{h}_5$ and $h_1$ are changed, see
Ref.~\cite{Nyffeler:2009uw}.

Finally, note that both VMD and LMD+V with $h_1 = 0$ can fit the data
sets A0, A1 and A2 for the transition form factor very well with
essentially the same $\chi^2$ per degree of freedom for a given data
set (see first and second part of the table).  Nevertheless, the
results for the pion-exchange contribution differ by about $20\%$ in
these two models. For VMD the result is $a_\mu^{{\rm LbyL}; \pi^0}
\sim 57.5 \times 10^{-11}$ and for LMD+V with $h_1 = 0$ it is $72.5
\times 10^{-11}$ with the JN approach and $80 \times 10^{-11}$ with
the MV approach. This is due to the different behavior, in these two
models, of the fully off-shell form factor ${\cal
  F}_{{\pi^0}^*\gamma^*\gamma^*}((q_1 + q_2)^2, q_1^2, q_2^2)$ on all
momentum variables, which enters for the pion-exchange
contribution~\cite{Jegerlehner_off-shell}. The VMD model is known to
have a wrong high-energy behavior with too strong damping, which
underestimates the contribution. For the VMD model, measurements of
the neutral pion decay width and the transition form factor completely
determine the model parameters $F_\pi$ and $M_V$ and the error given
in Table~\ref{tab:amu} is the total model error. Note that a smaller
error, compared to the off-shell LMD+V model, does not necessarily
imply that the VMD model is better, i.e.\ closer to reality. Maybe the
model is too simplistic.

We conclude that the KLOE-2 data with a total integrated luminosity of
$5$~fb$^{-1}$ will give a reasonable improvement in the part of the
$a_\mu^{{\rm LbyL}; \pi^0}$ error associated with the parameters
accessible via the $\pi^0 \to \gamma\gamma$ decay width and the
$\gamma^\ast\gamma \to \pi^0$ transition form factor. Depending on the
modelling of the off-shellness of the pion, there might be other,
potentially larger sources of uncertainty which cannot be improved by
the KLOE-2 measurements.

\section*{Acknowledgements}

I would like to thank my coauthors on Ref.~\cite{KLOE-2_impact} for
the pleasant collaboration on this work. I am grateful to the
organizers of MESON 2012 for the opportunity to present our paper and
for providing a stimulating atmosphere during the meeting. This work
is supported by funding from the Department of Atomic Energy,
Government of India, for the Regional Centre for Accelerator-based
Particle Physics (RECAPP), Harish-Chandra Research Institute.


\section{Introduction}
\subsection{Convex functions.} The Hermite-Hadamard inequality dates back to an 1883 observation of Hermite \cite{hermite} with an independent use by Hadamard \cite{hada} in 1893: it says that
for convex functions $f:[a,b] \rightarrow \mathbb{R}$
$$ \frac{1}{b-a} \int_{a}^{b}{f(x) dx} \leq \frac{f(a) + f(b)}{2}.$$
This inequality is rather elementary and has been refined in many ways -- we refer to the monograph of Dragomir \& Pearce \cite{drago}. However, there is relatively little work outside of the one-dimensional case; we refer to \cite{cal1, cal2, jianfeng, nicu2, nicu, choquet, past, stein1}.
 The strongest possible statement that one could hope for is, for convex functions $f:\Omega \rightarrow \mathbb{R}$ defined on convex domains $\Omega \subset \mathbb{R}^n$,
$$ \frac{1}{|\Omega|} \int_{\Omega}{f ~d x} \leq \frac{1}{|\partial \Omega|} \int_{\partial \Omega}{f ~d\sigma}.$$
This inequality has been shown to be true for many special cases: it is known for $\Omega = \mathbb{B}_3$ the $3-$dimensional ball by Dragomir \& Pearce \cite{drago} and $\Omega = \mathbb{B}_n$ by de la Cal \& Carcamo \cite{cal1} (other proofs are given by de la Cal, Carcamo \& Escauriaza \cite{cal2} and Pasteczka \cite{past}), the simplex \cite{bes}, the square \cite{square}, triangles \cite{chen} and Platonic solids \cite{past}.
It was pointed out by Pasteczka \cite{past} that if the inequality holds for a domain $\Omega$ with constant 1, then plugging in affine functions shows that the center of mass of $\Omega$ and the center of mass of $\partial \Omega$ coincide, which is not generally true for convex bodies; therefore the inequality cannot hold with constant 1 in higher dimensions uniformly over all convex bodies.
The first uniform estimate was shown in \cite{stein1}: if $f:\Omega \rightarrow \mathbb{R}$ is a convex, positive function on the convex domain $\Omega \subset \mathbb{R}^n$, then we have
\begin{equation} \label{full}
 \frac{1}{|\Omega|} \int_{\Omega}{f ~dx} \leq  \frac{c_n}{|\partial \Omega|} \int_{\partial \Omega}{f~d\sigma}
 \end{equation}
 with $c_n \leq 2 n^{n+1}$.
In this paper, we will improve this uniform estimate and show that the optimal constant satisfies $n-1 \leq c_n \leq 2 n^{3/2}$. We do not have a characterization of the extremal convex functions $f$ on a given domain $\Omega$ (however, see below, we have such a characterization in the larger family of subharmonic functions).

\subsection{Subharmonic functions.}
Niculescu \& Persson \cite{choquet} (see also \cite{cal2, nicu2}) have pointed out that one could also seek such inequalities for subharmonic functions, i.e. functions satisfying $\Delta f \geq 0$. We note that all convex functions are subharmonic. Jianfeng Lu and the last author \cite{stein1} showed that for all positive, subharmonic functions $f:\Omega \rightarrow \mathbb{R}$ on convex domains $\Omega \subset \mathbb{R}^n$
\begin{equation} \label{jianfeng}
\int_{\Omega}{f ~d x} \leq |\Omega|^{1/n} \int_{\partial \Omega}{f ~d \sigma}.
\end{equation}
Estimates relating the integral of a positive subharmonic function $f$ over $\Omega$ to the integral over the boundary $\partial \Omega$ are linked to the torsion function on $\Omega$ given by
\begin{align*}
- \Delta u &= 1 \quad \mbox{in}~\Omega\\
u &= 0 \quad \mbox{on}~\partial \Omega.
\end{align*}
Integration by parts and the inequalities $u \geq 0, \Delta f \geq 0$ show that
\begin{align*}
\int_{\Omega}{ f dx} = \int_{\Omega}{ f (-\Delta u) dx} &= \int_{\partial \Omega}{ \frac{\partial u}{\partial \nu} f d\sigma} - \int_{\Omega}{(\Delta f) u dx} \\
&\leq  \int_{\partial \Omega}{ \frac{\partial u}{ \partial \nu} f d\sigma} \\
&\leq \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)} \int_{\partial \Omega} { f d\sigma},
\end{align*}
where $\nu$ is the inward pointing normal vector.
This computation suggests that we may have the following characterization of the optimal constant for a given convex domain $\Omega$.
\begin{proposition}[see e.g. \cite{dragomirk, choquet}] The optimal constant $c(\Omega)$ in the inequality
$$\int_{\Omega}{f ~d x} \leq c(\Omega) \int_{\partial \Omega}{f ~d \sigma}$$
for positive subharmonic functions is given by
$$ c(\Omega) = \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)}.$$
\end{proposition}
The lower bound on $c(\Omega)$ follows from setting $f$ to be the Poisson extension of a Dirac measure located at the point at which the normal derivative assumes its maximum. The derivation also shows that it suffices to consider the case of harmonic functions $f$.
Implicitly, this also gives a characterization of extremizing functions (via the Green's function).
 Jianfeng Lu and the last author \cite{jianfeng} used this proposition in combination with a gradient estimate for the torsion function to show that the best constant in (\ref{jianfeng}) is uniformly bounded in the dimension.
We will follow a similar strategy to obtain an improved bound for the optimal constant in (\ref{jianfeng}).

\section{The Results}
Our first result improves the constant $c_n$ from (\ref{full}) in all dimensions for subharmonic functions and shows that the growth is at most polynomial.
\begin{theorem} Let $\Omega \subset \mathbb{R}^n$ be convex and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function. Then
\begin{equation} \label{thm1}
 \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},
 \end{equation}
where the optimal constant $c_n$ satisfies
$$ c_n \leq \begin{cases} n^{3/2}~\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mbox{if}~n~\mbox{is odd}, \\
\frac{n^2+n}{\sqrt{n+2}}  ~\qquad\mbox{if}~n~\mbox{is even}. \end{cases}$$
\end{theorem}

In particular, for $n=2$ dimensions, our proof shows the inequality
$$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{3}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},$$
where the constant 3 improves on constant 8 obtained earlier for convex functions in \cite{stein1}.
To complement the result in Theorem 1 we prove that any constant for which \eqref{thm1} is valid must grow at least linearly with the dimension.
\begin{theorem} The optimal constant $c_n$ in (\ref{thm1}) is non-decreasing in $n$ and satisfies
\begin{equation} \label{convex}
 c_n \geq \max\{n-1, 1\}.
 \end{equation}
\end{theorem}
In order to prove Theorem 2 we establish a connection to an isoperimetric problem that is of interest in its own right. Specifically, we prove the following Lemma.
\begin{lemma} In any dimension $n\geq 1$,
\begin{equation} \label{convex}
 \sup \left\{ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Omega_2|}: \Omega_2 \subset \Omega_1 ~ \emph{both convex domains in}~\mathbb{R}^{n}\right\}=n.
 \end{equation}
\end{lemma}
We are not aware of any prior treatment of this shape optimization problem in the literature.
Problem (\ref{convex}) can be equivalently written as
$$ \sup \left\{ \frac{|\partial \Omega|}{|\Omega|} \frac{1}{h(\Omega)}: \Omega ~ \mbox{a convex set in}~\mathbb{R}^{n-1}\right\},~ \mbox{where}~ h(\Omega) = \inf_{X \subset \overline{\Omega}}{ \frac{|\partial X|}{|X|}}$$
 denotes the Cheeger constant. We refer to Alter \& Caselles \cite{alter} and Kawohl \& Lachand-Robert \cite{kawohl}.
The result of Kawohl \& Lachand-Robert \cite{kawohl}
will be a crucial ingredient in the proof of Theorem 2 (we note that the infimum runs over all subsets, it is known that the Cheeger set is unique and convex).

We also obtain a slight improvement of the constant in (\ref{jianfeng}).

\begin{theorem} Let $\Omega \subset \mathbb{R}^n$ be convex and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function. Then
$$ \int_{\Omega}{f dx} \leq \frac{ |\Omega|^{1/n}}{ \omega_n^{1/n} \sqrt{n} } \int_{\partial \Omega}{ f d\sigma},$$
where $\omega_n$ is the volume of the unit ball in $n-$dimensions.
\end{theorem}
We observe that, as $n$ tends to infinity, $\omega_n^{1/n} \sqrt{n} \rightarrow \sqrt{2 \pi e}.$
We also note a construction from \cite{jianfeng} which shows that
the constant in Theorem 3 is at most a factor $\sqrt{2}$ from optimal in high dimensions.

\section{Proof of Theorem 1}
\subsection{Convex functions.}
We first give a proof of Theorem 1 under the assumption that $f$ is convex; this argument is fairly elementary and is perhaps useful in other settings. A full proof of Theorem 1 is given in \S 3.2.

\begin{proof}
This proof combines three different arguments. The first argument is that
\begin{equation} \label{b1}
  \int_{\Omega}{f dx} \leq \frac{w(\Omega)}{2} \int_{\partial \Omega}{ f d\sigma}
  \end{equation}
from the one-dimensional Hermite-Hadamard inequality applied along fibers that are orthogonal to the hyperplanes realizing the width $w(\Omega)$.

\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=1.6]
\draw [very thick] (0,0) -- (5,0);
\draw [dashed] (-0.5,0) -- (5.5,0);
\draw [dashed] (-0.5,1.05) -- (5.5,1.05);
\draw [very thick] (0,0) to[out=45, in=135] (5,0);
\draw [thick, <->] (6,0) -- (6,1);
\node at (6.3, 0.5) {$w(\Omega)$};
\node at (2.5, 0.5) {$\Omega$};
\draw [dashed] (1,0) -- (1,0.7);
\draw [dashed] (1.1,0) -- (1.1,0.7);
\draw [dashed] (1.2,0) -- (1.2,0.75);
\draw [dashed] (1.3,0) -- (1.3,0.8);
\end{tikzpicture}
\end{center}
\caption{Application of the one-dimensional inequality on a one-dimensional fiber. This step is lossy if the boundary is curved.}
\end{figure}

Steinhagen \cite{steinhagen} showed that width can be bounded in terms of the inradius
\begin{equation} \label{b2}
w(\Omega) \leq \begin{cases} 2 \sqrt{n} \cdot \mbox{inrad}(\Omega)~\,\,\,\,\,\qquad \mbox{if}~n~\mbox{is odd,} \\
2 \frac{n+1}{\sqrt{n+2}} \cdot\mbox{inrad}(\Omega) ~\qquad\mbox{if}~n~\mbox{is even}. \end{cases}
\end{equation}
The last inequality follows from \cite{larson1}: if $\Omega \subset \mathbb{R}^n$ is a convex body and
\begin{equation*}
	\Omega_t = \{x\in \Omega: d(x, \partial\Omega)>t\},
\end{equation*}
where $d(x, \partial \Omega)$ denotes the distance to the boundary
$$ d(x, \partial \Omega) = \inf_{y \in \partial \Omega}{ \|x-y\|},$$
then
$$|\partial \Omega_t| \geq |\partial \Omega| \left(1 - \frac{t}{\mbox{inrad}(\Omega)}\right)_+^{n-1}.$$
Since $|\nabla d(x, \partial \Omega)| = 1$ almost everywhere, the coarea formula implies
\begin{align*} |\Omega| &= \int_{0}^{ \tiny \mbox{inrad}(\Omega)} |\partial \Omega_t| dt \\
&\geq |\partial \Omega|\int_{0}^{ \tiny \mbox{inrad}(\Omega)}{ \left(1 - \frac{t}{\mbox{inrad}(\Omega)}\right)^{n-1} dt} = |\partial \Omega| \frac{\mbox{inrad}(\Omega)}{n}
\end{align*}
and thus we obtain (also stated in \cite[Eq. 13]{larson2})
\begin{equation} \label{b3}
\mbox{inrad}(\Omega) \leq n \frac{|\Omega|}{|\partial \Omega|}.
\end{equation}
Combining inequalities (\ref{b1}), (\ref{b2}) and (\ref{b3}) implies the result.
\end{proof}
Both Steinhagen's inequality as well as inequality (\ref{b3}) are sharp for the regular simplex. However, see Fig. 1, an application of the one-dimensional Hermite-Hadamard inequality can only be sharp if the fibers are hitting the boundary at a point at which they are normal, otherwise there is a Jacobian determinant determined by the slope of the boundary and better results are expected. It is not clear to us how to reconcile these two competing factors.

\subsection{A Proof of Theorem 1}
\begin{proof}  We have, for all positive, subharmonic functions $f:\Omega \rightarrow \mathbb{R}$,
$$ \int_{\Omega}{ f dx}  \leq \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)} \int_{\partial \Omega} { f d\sigma},$$
where $u$ is the torsion function.
A classic bound on the torsion functions is given in Sperb \cite[Eq. (6.12)]{sperb}),
$$  \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)} \leq \sqrt{2} \|u\|^{\frac{1}{2}}_{L^{\infty}}.$$
Moreover, using Steinhagen's inequality in combination with (\ref{b3}), we know that $\Omega$ is contained within a strip of width
$$ w(\Omega) \leq   \frac{|\Omega|}{|\partial \Omega|} \begin{cases} 2 n^{3/2} ~\,\,\,\,\,\qquad \mbox{if}~n~\mbox{is odd,} \\
2 \frac{n^2+n}{\sqrt{n+2}} ~\qquad\mbox{if}~n~\mbox{is even}. \end{cases}$$
We can now use the maximum principle to argue that the torsion function in $\Omega$ is bounded from above by the
torsion function in the strip of width $w(\Omega)$ (see Fig. 2). That torsion function, however, is easy to compute since the problem
becomes one-dimensional. Orienting the strip to be given by
$$ S = \left\{(x,y) \in \mathbb{R}^{n-1} \times \mathbb{R}: |y| \leq \frac{w(\Omega)}{2} \right\},$$
we see that the torsion function on the strip is given by
$$ v(x,y) = \frac{w(\Omega)^2}{8} - \frac{y^2}{2}.$$

\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=0.8]
\draw[very thick] (0,0) ellipse (3cm and 1cm);
\draw[thick] (-4, 1) -- (4, 1);
\draw[thick] (-4, -1) -- (4, -1);
\draw [<->] (5, -1) -- (5,1);
\node at (5.5,0) {$w(\Omega)$};
\node at (0,0) {$\Omega$};
\end{tikzpicture}
\end{center}
\caption{The torsion function in $\Omega$ is bounded from above by the torsion function of the strip.}
\end{figure}

This shows
$$\|u\|^{}_{L^{\infty}} \leq \frac{w(\Omega)^2}{8}$$
and thus
\begin{align*}
  \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)}  &\leq \sqrt{2} \|u\|_{L^{\infty}}^{1/2} \\
 &\leq \frac{w(\Omega)}{2}  \leq  \frac{|\Omega|}{|\partial \Omega|} \begin{cases}  n^{3/2} ~\,\,\,\,\,\qquad \mbox{if}~n~\mbox{is odd} \\
 \frac{n^2+n}{\sqrt{n+2}} ~\qquad\mbox{if}~n~\mbox{is even}. \end{cases}
\end{align*}
\end{proof}

\section{Proof of Theorem 2}

The purpose of this section is to prove $c_{n+1} \geq c_n$ as well as the inequality
$$  c_n \geq \sup \left\{ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Omega_2|}: \Omega_2 \subset \Omega_1 ~ \mbox{both convex domains in}~\mathbb{R}^{n-1}\right\}.$$
Theorem 2 is then implied by this statement together the proof of the Geometric Lemma in Section \S 5.

\begin{proof}
The proof is based on explicit constructions. We first show that $c_{n+1} \geq c_{n}$. This is straightforward and based on an extension in the $(n+1)-$first coordinate: for any $\varepsilon>0$, we can find a convex domain $\Omega_{\varepsilon} \subset \mathbb{R}^n$
and a positive, convex function $f_{\varepsilon}:\Omega_{\varepsilon} \rightarrow \mathbb{R}$ such that
$$ \frac{1}{|\Omega_{\varepsilon}|} \int_{\Omega_{\varepsilon}}{f_{\varepsilon}  dx} \geq \frac{c_n - \varepsilon}{ |\partial \Omega_{\varepsilon}| } \int_{\partial \Omega_{\varepsilon}}{ f_{\varepsilon}  d\sigma}.$$
We define, for any $z > 0$,
$$ \Omega_{z,\varepsilon} = \left\{(x, y): x \in \Omega_{\varepsilon} ~\mbox{and}~ 0 \leq y \leq z\right\} \subset \mathbb{R}^{n+1}$$
and $f_{z, \varepsilon}:\Omega_{z, \varepsilon} \rightarrow \mathbb{R}$ via
$$ f_{z, \varepsilon}(x,y) = f_{\varepsilon}(x).$$
Then
$$ \frac{1}{|\Omega_{z,\varepsilon}|} \int_{\Omega_{z,\varepsilon}}{f_{z,\varepsilon}  dx dy} =  \frac{1}{|\Omega_{\varepsilon}|} \int_{\Omega_{\varepsilon}}{f_{\varepsilon}  dx}.$$
This integral simplifies for $z$ large since
$$ \lim_{z \rightarrow \infty}  \frac{1}{ |\partial \Omega_{z,\varepsilon}| } \int_{\partial \Omega_{z,\varepsilon}}{ f_{z,\varepsilon}  d\sigma} =   \frac{1}{ |\partial \Omega_{\varepsilon}| } \int_{\partial \Omega_{\varepsilon}}{ f_{\varepsilon}  d\sigma}.$$
Picking $\varepsilon$ sufficiently small and $z$ sufficiently large shows that $c_{n+1} < c_n$ leads to a contradiction.\\

We now establish the inequality inequality
$$  c_n \geq \sup \left\{ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Omega_2|}: \Omega_2 \subset \Omega_1 ~ \mbox{both convex domains in}~\mathbb{R}^{n-1}\right\}.$$
To this end pick $0 \in \Omega_2 \subset \Omega_1 \subset \mathbb{R}^{n-1}$ in such a way that both domains are convex. We will now define a domain $\Omega_N \subset \mathbb{R}^n$ and a convex function $f_N:\Omega_N \rightarrow \mathbb{R}$ where $N \gg 1$ will be a large parameter. We first define the convex sets
$$ C_1 = \left\{(x, y): x \in \Omega_1 \,~\mbox{and}\,~ y \geq - N^3\right\}$$
and
$$ C_2 = \left\{(x,y): x \in \left(1- \frac{y}{N^2}\right)\Omega_2 \,~\mbox{and}~  y \leq N \right\}.$$
The set $\Omega_N$ is then given as the intersection $\Omega_N = C_1 \cap C_2$ (see Fig. 3). We observe that $\Omega_N$ is the intersection of two convex sets and is therefore convex. Also, looking at the scaling, we see that $C_1$ dominates: looking at $\Omega_N$ from `far away`, it looks essentially like $C_1$ truncated. We now make this precise: note that there exists a constant $\lambda \geq 1$ such that $\Omega_1 \subseteq \lambda \Omega_2$ and then
$$ \Omega_N \cap \left\{(x,y) \in \mathbb{R}^{n}: y \leq -(\lambda-1) N^2 \right\} =  C_1 \cap \left\{(x,y) \in \mathbb{R}^{n}: y \leq -(\lambda-1) N^2 \right\}.$$
This means, that for $N \gg \lambda$, the `left' part of the convex domain dominates area and volume.
We also observe that
\begin{align*}
 |\Omega_N| &= N^3 |\Omega_1| + \mathcal{O}(N^2) \\
 |\partial \Omega_N| &= N^3 |\partial \Omega_1| + \mathcal{O}(N^2),
 \end{align*}
 where the implicit constants depend on $\Omega_1$ and $\Omega_2$.
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=1.6]
\draw [<->, thick] (-1,0) -- (6,0);
\node at (6, -0.2) {$y$};
\draw[thick] (1,0) ellipse (0.25cm and 0.5cm);
\draw[thick] (0,0) ellipse (0.25cm and 0.5cm);
\filldraw (0,0) circle (0.02cm);
\node at (0, -0.15) {$-N^3$};
\draw [dashed] (0, 0.5) -- (5,0.5);
\draw [dashed] (0, -0.5) -- (5,-0.5);
\filldraw (4,0) circle (0.02cm);
\node at (4, -0.15) {0};
\draw[thick] (5,0) ellipse (0.2cm and 0.3cm);
\filldraw (5,0) circle (0.02cm);
\node at (5, -0.15) {$N$};
\draw[thick] (-1,0) ellipse (0.5cm and 1cm);
\draw[dashed] (-1, 1) -- (5, 0.3);
\draw[dashed] (-1, -1) -- (5, -0.3);
\end{tikzpicture}
\end{center}
\caption{The construction of $C_1$ and $C_2$.}
\end{figure}

Since $\Omega_2 \subset \Omega_1$, we have that
 $$ \Omega_N \cap \left\{(x,y) \in \mathbb{R}^{n}: y \geq 0 \right\} =  C_2 \cap \left\{(x,y) \in \mathbb{R}^{n}: y \geq 0 \right\}.$$
We now define a convex function on $\mathbb{R}^{n}$ via
$$ f(x,y) = \begin{cases} y \quad &\mbox{if}~y \geq 0, \\ 0 \quad &\mbox{otherwise.} \end{cases}$$
We obtain
$$ \int_{\Omega_N} f~ dx dy =  \int_{\Omega_N \cap \left\{y > 0\right\}} f~ dx dy=  \int_{C_2 \cap \left\{y > 0\right\}} f ~dx dy = (1+o(1)) \frac{N^2}{2} |\Omega_2|$$
$$ \int_{\partial \Omega_N} f~ d\sigma =   \int_{ \partial \Omega_N \cap \left\{y > 0\right\}} f~ d\sigma  = \int_{ \partial C_2 \cap \left\{y > 0\right\}} f ~d\sigma = (1+o(1)) \frac{N^2}{2} |\partial \Omega_2|.$$
This shows that
$$ \frac{1}{|\Omega_{N}|} \int_{\Omega_{N}}{f_{} ~ dx} = \frac{(1+o(1))}{2N} \frac{|\Omega_2|}{|\Omega_1|}$$
and
$$ \frac{1}{|\partial \Omega_{N}|} \int_{\partial \Omega_{N}}{f_{} ~ d\sigma} = \frac{(1+o(1))}{2N} \frac{|\partial \Omega_2|}{|\partial \Omega_1|}$$
which implies the desired result for $N \rightarrow \infty$.
\end{proof}

\section{Proof of the Geometric Lemma}

\begin{proof}
By the inequality
$$\frac{|\Omega|}{|\partial\Omega|}\leq \mbox{inrad}(\Omega)\leq n \frac{|\Omega|}{|\partial\Omega|},$$
 a proof of which can be found in~\cite{larson2}, the supremum is no larger than $n$ since
\begin{equation}\label{eq:upper bound}
	\frac{|\partial \Omega|}{|\Omega|} \frac{| \Omega'|}{|\partial \Omega'|}\leq \frac{n \cdot \mbox{inrad}(\Omega')}{\mbox{inrad}(\Omega)}\leq n.
\end{equation}
What remains is to prove that this upper bound is saturated.
The underlying idea of our proof is a theorem of Kawohl and Lachand-Robert \cite{kawohl} characterizing the Cheeger set of a convex set $\Omega\subset \mathbb{R}^2$. Specifically, their theorem states that for a convex $\Omega\subset \mathbb{R}^2$ the Cheeger problem
\begin{equation*}
	h(\Omega)=\inf\biggl\{\frac{|\partial\Omega'|}{|\Omega'|}: \Omega'\subset \overline{ \Omega}\biggr\}
\end{equation*}
is solved by the set
$$
	\Omega' = \{x\in \Omega: \exists y \in \Omega \mbox{ such that } x\in B_{1/h(\Omega)^{}}(y)\subset \Omega\},
$$
where $B_{r}(x_0)$ is a ball of radius $r$ centered at $x_0$.
We recall our use of the notation
\begin{equation*}
	\Omega_t = \{x\in \Omega: d(x, \partial\Omega)>t\},
\end{equation*}
where $d(x, \partial \Omega)$ denotes the distance to the boundary
$$ d(x, \partial \Omega) = \inf_{y \in \partial \Omega}{ \|x-y\|},$$
and we can equivalently write the Cheeger set of $\Omega$ as $\Omega'= \Omega_{1/h(\Omega)}+B_{1/h(\Omega)}$, where $B_r$ denotes a ball of radius $r$ centered in 0. Here and in what follows the sum of two sets is to be interpreted in the sense of the Minkowski sum:
\begin{equation*}
	A+B = \{x: x=a+b, a\in A, b\in B\}.
\end{equation*}

The situation when $n\geq 2$ is more complicated, and as far as we know a precise solution of the Cheeger problem is not available~\cite{alter}. Nonetheless, our aim in what follows is to prove that by taking $\Omega$ as a very thin $n$-simplex we can find a good enough candidate for $\Omega'$ among the one-parameter family of sets
\begin{equation}\label{eq:Kawohl_trialsest}
	   	\Omega_t + B_t, \quad 0\leq t\leq \mbox{inrad}(\Omega).
\end{equation}

We construct our candidate for $\Omega$ as follows. Let $\Omega(\eta)\subset \mathbb{R}^n$ be the $n$-simplex obtained by taking a regular $(n-1)$-simplex of sidelength $\eta\gg 1$ in the hyperplane $\{x \in \mathbb{R}^n:x_1=-1\}$ with $(-1, 0, \ldots, 0)$ as center of mass and adding the last vertex at $(h(\eta), 0, \ldots, 0)$, where $h(\eta)$ is chosen so that $\mbox{inrad}(\Omega(\eta))=1$. Note that, as $\eta$ becomes large, $h(\eta)$ is approximately $1$ and $|\Omega(\eta)|\sim \eta^{d-1}$. \\

By construction $B_1\subset \Omega(\eta)$, and it is the unique unit ball of maximal radius contained in $\Omega(\eta)$. Moreover, the set $\Omega(\eta)$ is a tangential body to this ball (that is, a convex body all of whose supporting hyperplanes are tangential to the same ball). Since every tangential body to a ball is homothetic to its form body~\cite{Schneider1} (in our case $\Omega(\eta)$ is in fact equal to its form body), the main result in~\cite{larson1} implies
\begin{equation*}
	|(\partial \Omega(\eta))_t| = (1-t)^{n-1}|\partial\Omega(\eta)|, \quad \mbox{ for all }t \in [0, 1].
\end{equation*}
An application of the coarea formula now yields the identity
\begin{equation}\label{eq:vol_per_ratio}
	|\Omega(\eta)|= \int_0^{1}|\partial(\Omega(\eta))_t|dt = \frac{|\partial\Omega(\eta)|}{n}.
\end{equation}
We also note that  $\Omega(\eta)\subset B_{2\eta}$. To see why this is true, we first note that the inradius of the regular $n$-simplex (by which we mean $n+1$ points all at distance 1 from each other embedded in $\mathbb{R}^n$) is given by
$$ r_n = \frac{1}{\sqrt{2n(n+1)}}.$$
The regular simplex is the convex body for which John's theorem is sharp, the circumradius is thus given by
$$ R_n = n\cdot r_n = \frac{\sqrt{n}}{\sqrt{2(n+1)}} \leq \frac{1}{\sqrt{2}}.$$
This shows that $\Omega(\eta) \subset B_{2\eta}$ (for the purpose of the proof, the constant 2 is not important and could be replaced by a much larger (absolute) constant).
Since it makes the computations somewhat simpler we consider, for a suitably chosen number $t$, the set $(1+t)\Omega(\eta)$. By construction $B_1\subset \Omega(\eta)$ which implies the inclusion
$$
	\Omega(\eta)+B_t\subset \Omega(\eta)+t\Omega(\eta)= (1+t)\Omega(\eta).
$$
In particular, we can test~\eqref{convex} with $\Omega=(1+t)\Omega(\eta)$ and $\Omega'=\Omega(\eta)+B_t$ for any values of $t, \eta\gg 1$.
We note that up to rescaling by $(1+t)^{-1}$ this is exactly the family of sets in~\eqref{eq:Kawohl_trialsest}. Indeed, for each $t$ the set $((1+t)\Omega(\eta))_t=\Omega(\eta)$.\\

The final step of the proof is to show that by letting $t, \eta \to \infty$ appropriately
\begin{equation}\label{eq:goal}
	\frac{|\partial((1+t)\Omega(\eta))|}{|(1+t)\Omega(\eta)|}\frac{|\Omega(\eta)+B_t|}{|\partial(\Omega(\eta)+B_t)|} \to n.
\end{equation}

To prove~\eqref{eq:goal} we recall the definition and some basic properties of mixed volumes~\cite[p. 275ff]{Schneider1}. Let $\mathcal{K}$ denote the set of convex bodies in $\mathbb{R}^n$ with nonempty interior. The mixed volume is defined as the unique symmetric function $W\colon \mathcal{K}^n \to \mathbb{R}_+$ satisfying
\begin{equation*}
  |\eta_1\Omega_1 + \ldots + \eta_m\Omega_m|= \sum_{j_1=1}^m\cdots \sum_{j_n=1}^m \eta_{j_1}\!\cdots \eta_{j_n}W(\Omega_{j_1}, \ldots, \Omega_{j_n}),
\end{equation*}
for any $\Omega_{1}, \ldots, \Omega_m \in \mathcal{K}$ and $\eta_1, \ldots, \eta_m\geq 0$~\cite{Schneider1}. Then $W$ satisfies the following properties:
\begin{enumerate}
  \item\label{itma} $W(\Omega_1, \ldots, \Omega_n)>0$ for $\Omega_1, \ldots, \Omega_n \in \mathcal{K}$.
  \item\label{itmb} $W$ is a multilinear function with respect to Minkowski addition.
  \item\label{itmc} $W$ is increasing with respect to inclusions in each of its arguments.

  \item\label{itme} The volume and perimeter of $\Omega\in \mathcal{K}$ can be written in terms of $W$:
  $$
    |\Omega|=W( \underbrace{\Omega, \ldots, \Omega}_{n~\mbox{\tiny times}}) \quad \mbox{and} \quad  |\partial\Omega|= n W(\underbrace{\Omega, \ldots, \Omega}_{n-1~\mbox{\tiny times}}, B_1).
  $$
\end{enumerate}
Since only mixed volumes of two distinct sets appear in our proof we introduce the shorthand notation
$$
	W_j(K, L) = W(\underbrace{K, \ldots, K}_{n-j}, \underbrace{L, \ldots, L}_{j}).
$$
By~\eqref{eq:vol_per_ratio}, we have
$$ 	\frac{|\partial((1+t)\Omega(\eta))|}{|(1+t)\Omega(\eta)|} = \frac{n}{1+t}.$$
The definition of $W$ implies
$$ |\Omega(\eta)+B_t| = W(\Omega(\eta)+B_t, \dots, \Omega(\eta)+B_t).$$
Multilinearity allows us to expand this term as
$$ W(\Omega(\eta)+B_t, \dots, \Omega(\eta)+B_t) = \sum_{j=0}^n \binom{n}{j}t^j W_j(\Omega(\eta), B_1)$$
and the same argument shows
$$ |\partial(\Omega(\eta)+B_t)| = n\sum_{j=0}^{n-1}\binom{n-1}{j}t^jW_{j+1}(\Omega(\eta), B_1).$$
Altogether, we can write the expression of interest as
\begin{equation}\label{eq:expansion in W}
	\frac{|\partial((1+t)\Omega(\eta))|}{|(1+t)\Omega(\eta)|}\frac{|\Omega(\eta)+B_t|}{|\partial(\Omega(\eta)+B_t)|}
	=
	\frac{n}{(1+t)}\frac{\sum_{j=0}^n \binom{n}{j}t^j W_j(\Omega(\eta), B_1)}{n\sum_{j=0}^{n-1}\binom{n-1}{j}t^jW_{j+1}(\Omega(\eta), B_1)}.
\end{equation}
In order to prove~\eqref{eq:goal} we need a bound from below. For the sum in the numerator it suffices  to keep the first two terms in the expansion and to use~Property (\ref{itma}) of $W$ resulting in
\begin{align*}
 	\sum_{j=0}^n \binom{n}{j}t^j W_j(\Omega(\eta), B_1)&\geq W_0(\Omega(\eta), B_1) + n t W_1(\Omega(\eta), B_1) \\
 	&= |\Omega(\eta)|+ t|\partial\Omega(\eta)|.
\end{align*}
To bound the sum in the denominator we wish to keep the term with $j=0$ as is.
For $j\geq 1$, we now use that $\Omega(\eta)\subset B_{2\eta}$ together with Property (\ref{itmc}) to bound
$$
	W_{j+1}(\Omega(\eta), B_1) \leq W_{j+1}(B_{2\eta}, B_{1})=(2\eta)^{n-j-1}|B_1|.
$$
Inserting the two bounds above into~\eqref{eq:expansion in W} yields
\begin{align*}
\frac{|\partial((1+t)\Omega(\eta))|}{|(1+t)\Omega(\eta)|}&\frac{|\Omega(\eta)+B_t|}{|\partial(\Omega(\eta)+B_t)|}\\
	&\geq
	\frac{n}{(1+t)}\frac{|\Omega(\eta)|+t|\partial\Omega(\eta)|}{|\partial\Omega(\eta)|+n\sum_{j=1}^{n-1}\binom{n-1}{j}2^{n-j-1}t^j\eta^{n-j-1} |B_1|}.
\end{align*}
We recall that $|\Omega(\eta)| = n^{-1} |\partial \Omega(\eta)|$ and therefore
\begin{align*}
\frac{|\partial((1+t)\Omega(\eta))|}{|(1+t)\Omega(\eta)|}&\frac{|\Omega(\eta)+B_t|}{|\partial(\Omega(\eta)+B_t)|}\\
	&\geq
	\frac{n t}{(1+t)}\frac{1 + 1/(nt)}{1+n|B_1|\sum_{j=1}^{n-1}\binom{n-1}{j}2^{n-j-1}t^j\eta^{n-j-1}/ |\partial\Omega(\eta)| }.
\end{align*}

By construction $|\partial\Omega(\eta)|\sim \eta^{n-1}$ as $\eta \to \infty$. Consequently, by choosing $t=\sqrt{\eta}$ (though the more general choice $t=\eta^{\alpha}$ for $0<\alpha<1$ would also work) we find $t^j \eta^{n-j-1}/|\partial\Omega(\eta)|\sim \eta^{-j/2}$. Therefore, taking $\eta$ (and thus $t$) to infinity, we obtain
\begin{equation*}
	\liminf_{\eta \to \infty}\frac{|\partial((1+\sqrt{\eta})\Omega(\eta))|}{|(1+\sqrt{\eta})\Omega(\eta)|}\frac{|\Omega(\eta)+B_{\sqrt{\eta}}|}{|\partial(\Omega(\eta)+B_{\sqrt{\eta}})|} \geq n,
\end{equation*}
which when combined with the matching upper bound~\eqref{eq:upper bound} completes the proof.
\end{proof}

\section{Proof of Theorem 3}
\begin{proof}
We use the estimate
$$ \int_{\Omega}{ f dx} \leq \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)} \int_{\partial \Omega} { f d\sigma}$$
introduced in Proposition 1 above and use, inspired by the argument in \cite{jianfeng}, estimates for the torsion function. One such estimate for the torsion function comes from $P-$functions, we refer to the classic book of Sperb \cite[Eq. (6.12)]{sperb},
$$  \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)} \leq \sqrt{2} \|u\|^{\frac{1}{2}}_{L^{\infty}}.$$
It remains to estimate the largest value of the torsion function. There are two different approaches: we can interpret it as the maximum lifetime of Brownian motion inside a domain of given measure or we can interpret it as the solution of a partial differential equation to which Talenti's theorem \cite{talenti} can be applied. In both cases, we end up with a standard isoperimetric estimate \cite{talenti} (that was also used in \cite{jianfeng})
$$ \|u\|_{L^{\infty}} \leq \frac{1}{2n}\left( \frac{|\Omega|}{\omega_n}\right)^{\frac{2}{n}}$$
to obtain
$$  \max_{x \in \partial \Omega}{  \frac{\partial u}{\partial \nu}(x)}  \leq \frac{ |\Omega|^{1/n}}{\omega_n^{1/n} \sqrt{n}}.$$
\end{proof}

\textbf{Acknowledgment.} This research was initiated at the workshop `Shape Optimization with Surface Interactions' at the American Institute of Mathematics in June 2019. The authors are grateful
to the organizers of the workshop as well as the Institute. KB's research was supported in part by Simons Foundation Grant 506732.
JL’s research was supported in part by a Bucknell University Scholarly Development Grant.
SL acknowledges financial support from Swedish Research Council Grant No. 2012-3864. SS's research was supported in part by the NSF (DMS-1763179) and the Alfred P. Sloan foundation.