\section{Introduction}

The Cauchy-Schwarz inequality on the Euclidean space $\mathbb{R}^{n}:$
\begin{equation}
\left( \sum_{i=1}^{n}v_{i}^{2}\right) \cdot \left(
\sum_{i=1}^{n}w_{i}^{2}\right) \geq \left( \sum_{i=1}^{n}v_{i}w_{i}\right)
^{2},  \label{1_0}
\end{equation
$\forall v=\mathbf{(}v_{1},...,v_{n}\mathbf{),}$ $w=\mathbf{(}w_{1},...,w_{n
\mathbf{)\in }\mathbb{R}^{n}$, is a basic result, with applications in
almost all the branches of mathematics.

In 1956, Acz\'{e}l \cite{AC} introduced the following inequality
\begin{equation}
\left( v_{0}^{2}-v_{1}^{2}-...-v_{n}^{2}\right) \left(
w_{0}^{2}-w_{1}^{2}-...-w_{n}^{2}\right) \leq \left(
v_{0}w_{0}-v_{1}w_{1}-...-v_{n}w_{n}\right) ^{2},  \label{1_1}
\end{equation
(holding for all $v\mathbf{=(}v_{0},v_{1},...,v_{n}\mathbf{),}$ $w\mathbf{=(
w_{0},w_{1},...,w_{n}\mathbf{)\in }\mathbb{R}^{n+1}$ such that
v_{0}^{2}-v_{1}^{2}-...-v_{n}^{2}>0,$ $w_{0}^{2}-w_{1}^{2}-...-w_{n}^{2}>0
), in relation to the theory of functional equations in one variable. The Ac
\'{e}l inequality (\ref{1_1}), together with its generalization to
Lorentzian manifolds, known by the name of \textit{reverse Cauchy-Schwarz
inequality,} proved to be crucial to relativity theory and to theories of
physical fields.

Indeed, from a geometric standpoint, the two inequalities above are known to
be two sides of the same coin; while the usual Cauchy-Schwarz inequality
\ref{1_0}) is extended to positive definite inner product spaces - and
further on, to Riemannian manifolds - leading to the triangle inequality
\left\Vert v+w\right\Vert \leq \left\Vert v\right\Vert +\left\Vert
w\right\Vert ,$ (\ref{1_1}) is naturally extended to spaces with a
Lorentzian inner product (and more generally, to Lorentzian manifolds),
leading to a reverse triangle inequality (see, e.g., \cite{BeemEhrlich,ONeil
).

\bigskip

In this article we discuss a further generalization of the above picture
which has not been exploited so far. The Cauchy-Schwarz inequality and its
Lorentzian-reversed version can be extended to Finsler, \cite{Bao},
respectively, to Lorentz-Finsler spaces, \cit
{Aazami:2014ata,Javaloyes2019,Minguzzi2014,Minguzzi:2013sxa}. Roughly
speaking, while a Riemannian manifold is a space equipped with a smoothly
varying family of inner products, a Finsler manifold is a space equipped
with a family of \textit{norms}\footnote
Actually, the notion of Finsler norm is slightly more general than the usual
one, as it is only required to be positively homogeneous intead of
absolutely homogeneous.} that do not necessarily arise from a scalar
product. Similarly, a Lorentz-Finsler manifold is equipped with a smoothly
varying family of so-called \textit{Lorentz-Finsler norms} of vectors, that
do not necessarily arise as the square root of any quadratic expression -
but are just positively 1-homogeneous in the considered vectors.

\bigskip

Usually, the Finslerian Cauchy-Schwarz inequality (also called the \textit
fundamental inequality, }\cite{Bao}) is proven under the assumption that the
Finsler norm $F$ of vectors has the property that the Hessian $Hess(F^{2})$
is positive definite; in the particular case of Riemannian spaces, this will
turn into the condition that the metric tensor $g$ is positive definite.
Similarly, its Lorentzian-reversed counterpart, \cit
{Aazami:2014ata,Javaloyes2019,Minguzzi2014,Minguzzi:2013sxa} is proven under
the assumption that $Hess(F^{2})$ has Lorentzian signature for all vectors
in a strictly convex set. Under these assumptions, the obtained inequalities
are \textit{strict}, i.e., equality only holds when the vectors $v$ and $w$
are collinear.

\bigskip

As a preliminary step, we show that the above conditions can be relaxed.
Namely, the respective inequalities still hold - just, non-strictly - if we
allow $Hess(F^{2})$ to be degenerate along some directions; also, for
practical matters, we replace the strict convexity assumption on the set of
interest with the more relaxed one that $F$ is defined on a convex conic
domain\textbf{\ }$\mathcal{T}$. Moreover, we prove two refinements of the
reverse triangle inequality holding in general Lorentz-Finsler spaces in
Section \ref{Section degenerate case}.

\bigskip

While the above generalization is not spectacular in itself, it allows us
much more freedom in choosing the range of examples and applications.
Indeed, we show in Section~\ref{sec:ex} that some of the most famous
inequalities on $\mathbb{R}^{n}$ are nothing but reverse Cauchy-Schwarz
inequalities for conveniently chosen (possibly, degenerate) Lorentz-Finsler
norms:

\begin{enumerate}
\item The usual arithmetic-geometric mean inequality: $\dfrac{1}{n}~\overset
n}{\underset{i=1}{\dsum }}v_{i}\geq {\large (}\overset{n}{\underset{i=1}
\dprod }}v_{i}{\large )}^{1/n},$ $\ \ \forall v_{i}\geq 0,$ $i=\overline{1,n
.$

\item The weighted arithmetic-geometric mean inequality:
\begin{equation}
\dfrac{1}{a}\sum_{i=1}^{n}a_{i}v_{i}\geq \lbrack
(v_{1})^{a_{1}}...(v_{n})^{a_{n}}]^{1/a},  \label{weighted_a}
\end{equation
for all $a,a_{i},v_{i}\in \mathbb{R}_{+}^{\ast }$, such that $\overset{n}
\underset{i=1}{\sum }}a_{i}=a.$

\item Popoviciu's inequality, \cite{Pop}:
\begin{equation}
\left( v_{0}^{p}-v_{1}^{p}-...-v_{n}^{p}\right) ^{1/p}\left(
w_{0}^{q}-w_{1}^{q}-...-w_{n}^{q}\right) ^{1/q}\leq
v_{0}w_{0}-v_{1}w_{1}-...-v_{n}w_{n},  \label{1_2}
\end{equation
holding for all $v_{i},w_{i}>0,$ such that
v_{0}^{p}-v_{1}^{p}-...-v_{n}^{p}>0$ and
w_{0}^{q}-w_{1}^{q}-...-w_{n}^{q}>0;$ the powers $p,q>1$ are such that
\dfrac{1}{p}+\dfrac{1}{q}=1$. In particular, for $p=q=\dfrac{1}{2},$
Popoviciu's inequality yields Acz\'{e}l's inequality.

\item Another result, due to Bellman \cite{Bel, Mitrinovic1970,
Mitrinovic1993}:
\begin{equation}
\left( v_{0}^{p}-v_{1}^{p}-...-v_{n}^{p}\right) ^{1/p}+\left(
w_{0}^{p}-w_{1}^{p}-...-w_{n}^{p}\right) ^{1/p}\leq \lbrack
(v_{0}+w_{0})^{p}-(v_{1}+w_{1})^{p}-...-(v_{n}+w_{n})^{p}]^{1/p},
\label{Bellman}
\end{equation
(with $v_{i},w_{i}$ as above and$\ p>1$) is just the reverse triangle
inequality corresponding to (\ref{1_2}). \newline
\end{enumerate}

Further, we use two particular classes of Lorentz-Finsler norms (the
so-called \textit{bimetric} and \textit{Kropina} norms) in order to prove
some new inequalities in Sections~\ref{Section_bimetric} and \re
{Section_Kropina}.

In Section \ref{Aczel section}, we study the following class of inequalities
on $\mathbb{R}^{n+1}$
\begin{equation}
\lbrack v_{0}w_{0}-\bar{g}_{\vec{v}}(\vec{v},\vec{w})]^{2}-[v_{0}^{2}-\lef
\Vert \vec{v}\right\Vert ^{2}][w_{0}^{2}-\left\Vert \vec{w}\right\Vert
^{2}]\geq 0  \label{Aczel_Finsler}
\end{equation
(for all $\vec{v}=\left( v_{1},....,v_{n}\right) ,\vec{w}=\left(
w_{1},....,w_{n}\right) \in \mathbb{R}^{n},$ $v_{0},w_{0}>0$ such that
v_{0}^{2}-\left\Vert \vec{v}\right\Vert ^{2}\geq 0,$ $w_{0}^{2}-\left\Vert
\vec{w}\right\Vert ^{2}\geq 0$), where $\left\Vert \vec{v}\right\Vert =\bar{
}(\vec{v})$ is an arbitrary Finsler norm on $\mathbb{R}^{n}$ and $\bar{g}_
\vec{v}}=\dfrac{1}{2}Hess_{\vec{v}}(\bar{F}^{2})$ is the corresponding
Finsler metric tensor -- thus generalizing the usual Acz\'{e}l inequality.
Using the positive definite version of the Finslerian Cauchy-Schwarz
inequality, we then find two refinements thereof:

\begin{equation}
\lbrack v_{0}w_{0}-\bar{g}_{\vec{v}}(\vec{v},\vec{w})]^{2}-[v_{0}^{2}-\lef
\Vert \vec{v}\right\Vert ^{2}][w_{0}^{2}-\left\Vert \vec{w}\right\Vert
^{2}]\geq \dfrac{\left( w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2
}{\left\Vert \vec{w}\right\Vert ^{2}}\left( \left\Vert \vec{v}\right\Vert
^{2}\left\Vert \vec{w}\right\Vert ^{2}-\bar{g}_{\vec{v}}(\vec{v},\vec{w
)\right) ;  \label{refinement1}
\end{equation}

\begin{equation}
\lbrack v_{0}w_{0}-\bar{g}_{\vec{v}}(\vec{v},\vec{w})]^{2}-[v_{0}^{2}-\lef
\Vert \vec{v}\right\Vert ^{2}][w_{0}^{2}-\left\Vert \vec{w}\right\Vert
^{2}]\geq \left[ w^{0}\dfrac{\bar{g}_{\vec{v}}(\vec{v},\vec{w})}{\left\Vert
\vec{w}\right\Vert }-v^{0}\left\Vert \vec{w}\right\Vert \right] ^{2}.
\label{refinement2}
\end{equation}

\section{Reverse inequalities for Lorentzian bilinear forms\labe
{sec:classineq}}

Before we study the extended Finslerian case, we briefly recall the
classical inequalities for bilinear forms.

Throughout the paper, we denote by $V$ a real $\left( n+1\right)
-dimensional space. We will use Einstein's summation convention, if not
otherwise explicitly stated: whenever in an expression an index $i$ appears
both as a superscript and as a subscript, we will automatically understand
summation over all possible values of $i$, i.e., instead of $\underset{i=0}
\overset{n}{\sum }}a_{i}b^{i},$ we will write simply, $a_{i}b^{i}$. This is
why, we will typically number components of vectors with superscripts from
0 $ to $n$, rather than as subscripts; this way, the expression of a vector
v\in V$ in the basis $\left\{ e_{i}\right\} _{i=\overline{0,n}}$ will be
written a
\begin{equation*}
v=v^{i}e_{i}.
\end{equation*
Unless elsewhere specified, by "smooth", we will always mean $\mathcal{C
^{\infty }$ (though usually, differentiability of some finite order is
sufficient). We will denote by $i,j,k...$\ indices running from $0$\ to $n$\
and by Greek letters $\alpha ,\beta ,\gamma ,..,$\ indices running from $1$\
to $n.$

\bigskip

A \textit{Lorentzian scalar product, \cite{BeemEhrlich, Minguzzi2019, ONeil}
} on the $\left( n+1\right) $-dimensional vector space $V$ is a symmetric
bilinear form $g:V\times V\rightarrow \mathbb{R}$ of index $n$. If $V$
admits a Lorentzian scalar product, it is called an $(n+1)$-dimensional
\textit{Minkowski spacetime}. Choosing an arbitrary basis, we have:
\begin{equation}
g(v,v)=g_{ij}v^{i}v^{j},  \label{Lorentzian bilinear form general}
\end{equation
where $(g_{ij})$ is a matrix with constant entries. In particular, in a $g
-orthonormal basis of $V,$ the bilinear form $g$ has the expressio
\begin{equation}
g(v,v)=\eta _{ij}v^{i}v^{j}=\left( v^{0}\right) ^{2}-\left( v^{1}\right)
^{2}-...-\left( v^{n}\right) ^{2},  \label{Minkowski metric}
\end{equation
where $\left( \eta _{ij}\right) =diag(1,-1,-1,...,-1).$

A nonzero vector $v\in V$ is called \textit{timelike} if $g(v,v)>0$ and
\textit{causal}, if $g(v,v)\geq 0$. The set of causal vectors consists of
two connected components, corresponding to the choices $v^{0}>0$ and
v^{0}<0 $ respectively in a given (arbitrary) $g$-orthonormal basis.

In the following, we will denote by $C$ one of these two connected
components. By conveniently choosing the basis, we can assume that, for all
v\in C,$ we have $v^{0}>0.$ The elements of $C$ are called \textit
future-directed causal vectors.}

Denote
\begin{equation}
F(v):=\sqrt{g(v,v)},~\ \ \forall v\in C.  \label{pseudo-norm}
\end{equation
The function $F:C\rightarrow \mathbb{R}^{+}$ defined by the above relation
is sometimes called, by analogy with the Euclidean case, the Lorentzian
(pseudo-)\textit{norm }associated to the Lorentzian scalar-product $g$.

We will denote by
\begin{equation}
\mathcal{T}~\ \mathcal{=}\left\{ v\in C~\ |~F(v)>0\right\} ,
\label{definition_T}
\end{equation
the subset of $C$ consisting of timelike vectors. Elements of $\mathcal{T}$
are called \textit{future-directed timelike vectors}. The set $\mathcal{T}$
is always convex.

On a Minkowski spacetime $(V,g),$ the following inequalities hold (see,
e.g., \textit{\cite[Proposition 30]{ONeil}})

\begin{itemize}
\item \textbf{Reverse Cauchy-Schwarz inequality}
\begin{equation}
g(v,w)\geq F(v)F(w),~\ \ \ \forall v,w\in C;
\label{classical reverse CS ineq}
\end{equation}

\item \textbf{Reverse triangle inequality:
\begin{equation*}
F(v+w)\geq F(v)+F(w),~\ \forall v,w\in C.
\end{equation*
These inequalities are \textit{strict}, in the sense that equality holds if
and only if $v$ and $w$ are collinear.
\end{itemize}

\bigskip

\textbf{Particular case (Acz\'{e}l's inequality). }For $V=\mathbb{R}^{n+1}$
equipped with a $g$-orthonormal basis, the reverse Cauchy-Schwarz inequality
\begin{equation}
(v^{0}w^{0}-v^{1}w^{1}-....-v^{n}w^{n})^{2}\geq \lbrack \left( v^{0}\right)
^{2}-\left( v^{1}\right) ^{2}...-\left( v^{n}\right) ^{2}][\left(
w^{0}\right) ^{2}-\left( w^{1}\right) ^{2}...-\left( w^{n}\right) ^{2}],
\label{Aczel}
\end{equation
$\forall v,w\in C$, becomes Acz\'{e}l's inequality (\ref{1_1}).

\bigskip

\begin{remark}
\textbf{(Positive definite bilinear form\textbf{s}):} In the case when the
metric $g$ is positive definite, we have: $C=V,$ $\mathcal{T}=V\backslash
\{0\}.$ The usual, non-reversed Cauchy-Schwarz inequality
\begin{equation}
g(v,w)\leq F(v)F(w)  \label{non-reversed CS}
\end{equation
and the usual triangle inequality
\begin{equation}
F(v+w)\leq F(v)+F(w)  \label{non-reversed triangle ineq}
\end{equation
hold strictly on the entire space $V,$ see for example \cite[Proposition 18
{ONeil}.
\end{remark}

\section{\label{ineq_Finsler norms}Finsler and Lorentz-Finsler functions on
a vector space}

We saw in the previous section that the famous reverse and non-reverse
triangle inequalities, thus in particular Acz\'{e}l's inequality, are
closely connected to the geometric concepts of (pseudo)-Riemannian geometry.
Here we present that further famous inequalities are also related to a
geometric concept, namely to the concept of \textit{(pseudo)-Finsler geometr
}.

\subsection{Finsler structures on a vector space}

Let $V$ be a real $\left( n+1\right) $-dimensional space as above.

A \textit{Finsler} norm on $V$ is "almost" a norm in the usual sense; the
difference consists in the fact that it is only \textit{positively
homogeneous, instead of absolutely homogeneous. The precise definition is
given below.

\begin{definition}
(\cite{Bao}): A Finsler norm on the vector space $V$ is a function
F:V\rightarrow \lbrack 0,\infty )$ with the following properties:

1) $F$ is smooth on $\mathcal{T}${\textbf{$=$}}$V\backslash \{0\}$ and
continuous at $v=0;$

2) $F$ is positively homogeneous of degree 1, i.e., $F(\lambda v)=\lambda
F(v),$ $\forall \lambda >0;$

3) For every $v\in \mathcal{T},$ the fundamental tensor $g_{v}:V\times
V\rightarrow \mathbb{R},$
\begin{equation}
\ g_{v}(u,w):=\dfrac{1}{2}\dfrac{\partial ^{2}F^{2}}{\partial t\partial s
(v+tu+ws)\mid _{t=s=0}  \label{g_ definition}
\end{equation
is positive definite.
\end{definition}

\bigskip

\textbf{Note: }In the Finsler geometry literature, a function $F$ with the
above properties is called a \textit{Minkwoski norm}. Yet, in order to avoid
confusions with the \textit{Minkowski metric} $\eta $ as defined above, we
will avoid this terminology here and call $F$ instead, a \textsl{Finsler
\textit{\ }norm.

\bigskip

\textbf{Particular case. }Euclidean spaces are recovered for $F(v)=\sqrt
a_{ij}v^{i}v^{j}},$ where $\left( a_{ij}\right) $ is a constant matrix
(i.e., $g_{v}=a$ does not depend on $v$). In this case, the Finsler norm $F$
is a Euclidean one, since it arises from a scalar product.

\bigskip

Yet, generally, a Finsler norm does not generally arise from a scalar
product. Nevertheless, there exists a similar notion to a scalar product -
namely, the \textit{fundamental }(or \textit{metric})\textit{\ tensor }$g_{v}
$ - but, in general $g_{v}$ has a nontrivial dependence on the vector $v.$
More precisely, the fundamental tensor of the Finsler space $(V,F)$ is the
mapping $g:V\backslash \{0\}\rightarrow T_{2}^{0}(V),$ $v\mapsto g_{v},$
which associates to each vector $v$ the symmetric and positive definite
bilinear form $g_{v.}$ defined above. With respect to an arbitrary basis
\left\{ e_{i}\right\} _{i=\overline{0,n}}$ of $V,$ the fundamental tensor
g_{v}$ has the matrix
\begin{equation}
g_{ij}(v)=\dfrac{1}{2}\dfrac{\partial ^{2}F^{2}}{\partial v^{i}\partial v^{j
}(v).  \label{metric tensor}
\end{equation
that is
\begin{equation}
g_{v}(u,w)=g_{ij}(v)u^{i}w^{j}.  \label{g_v bilinear}
\end{equation
Hence, for each $v\in V,~g_{v}$ is a scalar product (with "reference vector"
$v$) on $V.$ Moreover, due to the homogeneity of $F,$ there holds a similar
formula to the one in Euclidean geometry:
\begin{equation}
F(v)=\sqrt{g_{v}(v,v)}.  \label{F in terms of g_v}
\end{equation}

\bigskip

In the following, we denote the derivatives of $F$ with subscripts:\ $F_{i}:
\dfrac{\partial F}{\partial v^{i}},$ $F_{ij}=\dfrac{\partial ^{2}F}{\partial
v^{i}\partial v^{j}}$ etc.

At any $v\in V\backslash \{0\},$ the Hessian of $F:$
\begin{equation}
F_{ij}(v)=\dfrac{1}{F}[g_{ij}(v)-F_{i}(v)F_{j}(v)],  \label{angular metric}
\end{equation
is positive semidefinite, with radical spanned by $v.$ This fact serves to
prove, (see \cite{Bao}, p. 8-9):

1)\ \textit{the fundamental (or Cauchy-Schwarz) inequality:
\begin{equation}
dF_{v}(w)\leq F(w),~\ \forall v,w\in V\backslash \{0\};
\label{pos def CS Finsler}
\end{equation}

2)\ \textit{the triangle inequality:
\begin{equation}
F(v+w)\leq F(v)+F(w),~~\forall v,w\in V.
\label{pos def triangle ineq Finsler}
\end{equation
The above inequalities are strict, i.e., equality only holds when $v$ and $w$
are collinear.

\bigskip

With respect to a given basis, the fundamental inequality takes the form
\begin{equation}
F_{i}(v)w^{i}\leq F(w).  \label{pos_def_CS_FInsler_coords}
\end{equation
The name of Cauchy-Schwarz inequality for (\ref{pos_def_CS_FInsler_coords})\
is justified by the following. Noticing tha
\begin{equation}
dF_{v}(w)=F_{i}(v)w^{i}=\dfrac{g_{ij}(v)v^{j}w^{i}}{F(v)}=\dfrac{g_{v}(v,w)}
F(v)},  \label{F_w}
\end{equation
this inequality can be equivalently written as
\begin{equation}
g_{v}(v,w)\leq F(v)F(w),  \label{pos def CS Finsler detailed}
\end{equation
i.e., the fundamental inequality (\ref{pos def CS Finsler}) is just a
generalization of the usual Cauchy-Schwarz inequality \eqref{non-reversed CS
.

\subsection{Lorentz-Finsler structures}

An important feature of Lorentz-Finsler functions is that, typically, they
can only be defined on a conic subset of $V.$

In the following, by\textbf{\ }a \textit{conic domain} of $V,$ we will mean
an open connected subset $\mathcal{Q}$ of $V\backslash \{0\}$ with the
\textit{conic property}:
\begin{equation*}
\forall v\in \mathcal{Q},\forall \lambda >0:~\lambda v\in \mathcal{Q}.
\end{equation*}

The definition below is slightly more general than the one by Javaloyes and
Sanchez, \cite[ p. 21]{Javaloyes2019}:

\begin{definition}
Let $\mathcal{T}\subset V\backslash \{0\}$ be a conic domain. We call a
Lorentz-Finsler norm on $\mathcal{T}$ a smooth function $F:\mathcal{T
\rightarrow (0,\infty )$ such that:

1) $F$ is positively homogeneous of degree 1: $F(\lambda v)=\lambda F(v),$
\forall \lambda >0,$ $\forall v\in \mathcal{T}.$

2) For every $v\in \mathcal{T},$ the fundamental tensor $g_{v}:V\times
V\rightarrow \mathbb{R},$
\begin{equation*}
\ g_{v}(u,w):=\dfrac{1}{2}\dfrac{\partial ^{2}F^{2}}{\partial t\partial s
(v+tu+ws)\mid _{t=s=0}
\end{equation*
has Lorentzian signature $(+,-,-,...,-)$.
\end{definition}

A Lorentz-Finsler norm can always be prolonged as 0 at $v=0.$

\bigskip

\textbf{Notes:}

1)\ The difference between the above introduced notion and the one of
\textit{Lorentz-Minkowski norm} presented in \cite{Javaloyes2019} is that we
will \textit{not} require $F$ to be extended as 0 on $\partial \mathcal{T};$
while this requirement is important to applications in physical theories, in
our case, it would just uselessly limit the range of allowed examples (see,
e.g., Section \ref{Section_Popoviciu}). Actually, as we will see in the next
section, we will even allow $F$ to be degenerate at some vectors $v\in
\mathcal{T}$.

2) Equipping a differentiable manifold $M$ with a smooth family of Lorentz
Finsler norms $p\mapsto F(p)$ which define a Lorentz Finsler structure $F(p)$
on each tangent space $T_{p}M,$ $p\in M$, and demanding that $F|_{\partial
\mathcal{T}}=0,$ makes the pair $(M,L=F^{2})$ a Finsler spacetime, \cit
{cosmo-Berwald,Javaloyes2019}. Finsler spacetimes gain attention in the
application to gravitational physics \cite{Hohmann:2018rpp,Pfeifer:2019wus},
as well as in the mathematical community as generalizations of Lorentzian
manifolds \cite{Bernal:2020bul}.

3)\ If, in the above definition, one replaces the condition of Lorentzian
signature with positive definiteness, one obtains the notion of (positive
definite) \textit{conic Finsler metric, \cite{Javaloyes2019}}. Thus, a usual
Finsler metric is a conic Finsler metric with $\mathcal{T}=V\backslash
\{0\}. $

\bigskip

For Lorentz-Finsler norms $F$, the matrix $g_{ij}(v)$ is defined by the same
formula (\ref{metric tensor}) (but this time, it has Lorentzian signature)
and the relation $F(v)=\sqrt{g_{v}(v,v)}$ still holds. The Hessian $F_{ij}$
is negative semidefinite with radical spanned by $v,$ i.e.
\begin{equation}
F_{ij}(v)w^{i}w^{j}\leq 0,  \label{Hess(F)_ineq}
\end{equation
for all $v,w\in \mathcal{T},$ where equality implies that $w$ is collinear
to $v$. Conversely, if the radical of $(F_{ij}(v))$\ is 1-dimensional, then
g_{v}\ $has $(+,-,-,...,-)$ signature, (see\footnote
The proof of (\ref{Hess(F)_ineq}) inside the open conic set $\mathcal{T}$ in
the cited paper does not require $F$ to be extendable on $\partial T,$ hence
the result holds with no modification in our case\textbf{.}} \cit
{Javaloyes2019}, Proposition 4.8 and, respectively, Lemma 4.7)\textbf{.}

\bigskip

\textbf{Examples of Lorentz-Finsler norms. }Here we just briefly list some
examples $F:\mathcal{T}\rightarrow \mathbb{R}$ (defined on conic subsets
\mathcal{T\subset }\mathbb{R}^{n+1}$), to be examined in the following
sections.

1) The $(n+1)$-dimensional \textit{Minkowski metric}: $F(v)=\sqrt{\eta
_{ij}v^{i}v^{j}}.$

2)\ \textit{The Kropina spacetime metric: }$F(v)=\dfrac{\eta _{ij}v^{i}v^{j
}{v^{0}}$.

3) \textit{The }$p$\textit{-pseudo-norm}: $F(v)=$ $\left[ \left(
v^{0}\right) ^{p}-\left( v^{1}\right) ^{p}-...-\left( v^{n}\right) ^{p
\right] ^{\tfrac{1}{p}}.$

4)\ The $(n+1)$-dimensional Berwald-Mo\'or metric
F(v)=(v^{0}v^{1}...v^{n})^{\tfrac{1}{n+1}}.$

5)\ Bimetric spaces: $~F(v)=[(\eta _{ij}v^{i}v^{j})(h_{kl}v^{k}v^{l})]^
\tfrac{1}{4}}$, where $h_{kl}v^{k}v^{l}$ has Lorentzian signature.

The latter three examples belong to a wider class of Lorentz-Finsler
functions $F$, called $m$\textit{-th root metrics}, expressed as the $m$-th
root of some polynomial of degree $m>2$ in $v^{i}$.

\subsection{The degenerate case\label{Section degenerate case}}

In previous works on the topic, such as \cit
{Aazami:2014ata,Javaloyes2019,Minguzzi2014,Minguzzi:2013sxa}, the Finslerian
generalizations of the reverse Cauchy-Schwarz inequality and of the reverse
triangle inequality were proven under the hypothesis that the set $B(1)
\mathcal{T}\cap F^{-1}([1,\infty ))$ is strictly convex (a sufficient
condition thereof is that $F$ vanishes on $\partial \mathcal{T}$ - which, as
we mentioned above, is not assumed here). These inequalities are strict,
i.e., equality happens if and only if $v$ and $w$ are collinear. Also, in
\cite{Javaloyes2019}, it is proven that, if $B(1)$\ is just (non-strictly)\
convex, then the inequalities hold non-strictly.

\bigskip

In the following, we will prove that these inequalities still hold if we
relax both the nondegeneracy condition on $F$ and the convexity requirement;
while requiring that the open cone $\mathcal{T}$ is (non-strictly) convex
does not affect the strictness of the inequalities, allowing $F\ $to be
degenerate does indeed affect it. With this goal in mind, let us first prove
a lemma, which extends (\ref{Hess(F)_ineq}) to degenerate Finsler structures.

\begin{lemma}
\label{signature_Hess(F)} Consider a smooth, 1-homogeneous function $F
\mathcal{T\rightarrow }\mathbb{R}$ defined on an arbitrary conic domain
\mathcal{T}\subset V\backslash \{0\}$ and denote, with respect to an
arbitrary basis:
\begin{equation*}
g_{ij}(v)=\dfrac{1}{2}\dfrac{\partial ^{2}F^{2}}{\partial v^{i}\partial v^{j
}(v).
\end{equation*
Then:

(i) If, at some $v\in \mathcal{T},$ the matrix $g_{ij}(v)$ has only one
positive eigenvalue, then the Hessian $(F_{ij}(v))$ is negative semidefinite.

(ii) If, at some $v\in \mathcal{T},$ the matrix $g_{ij}(v)$ is positive
semidefinite, then the Hessian $(F_{ij}(v))$ is positive semidefinite.
\end{lemma}

\begin{proof}
\textit{(i) }Fix $v\in \mathcal{T}$. Then, using
g_{ij}(v)=FF_{ij}(v)+F_{i}(v)F_{j}(v),$ we find for any $u\in V:
\begin{equation}
FF_{ij}(v)u^{i}u^{j}=g_{ij}(v)u^{i}u^{j}-(F_{i}(v)u^{i})^{2}.
\label{ineq_F_g}
\end{equation
As the signature of $g_{v}$ does not depend on the choice of the basis
\left\{ e_{i}\right\} _{i=\overline{0,n}}$, we can freely choose this basis.
For instance, we can choose an orthogonal basis for $g_{v},$\ with $e_{0}=v.$
Since $g_{v}(e_{0},e_{0})=g_{ij}(v)v^{i}v^{j}=F^{2}(v)>0,$ it follows from
the hypothesis that all the other diagonal entries $g_{ii}(v)$ are
nonpositive. Setting $u=e_{\alpha }$ for $\alpha \not=0,$ the orthogonality
condition is written, taking into account (\ref{F_w}), as $F_{i}(v)u^{i}=0;$
therefore, $F_{ij}(v)u^{i}u^{j}=\dfrac{1}{F(v)}g_{ij}(v)u^{i}u^{j}\leq 0.$
But, on the other hand, we have:
F_{ij}(v)e_{0}^{i}e_{0}^{j}=F_{ij}(v)v^{i}v^{j}=0,$ i.e, evaluating the
bilinear form $F_{ij}(v)$ on any basis vector $e_{\alpha }$, we get
nonpositive values, i.e., $F_{ij}(v)$ is negative semidefinite for any $v\in
\mathcal{T}$.

\textit{(ii) }is proven similarly, taking into account that, this time,
F_{ij}(v)u^{i}u^{j}=g_{ij}(v)u^{i}u^{j}\geq 0.$
\end{proof}

We obtain two statements:

\begin{theorem}
\label{prop_degenerate Finsler case}\textbf{(The degenerate-Lorentzian case)
} Let $\mathcal{T}\subset V\backslash \{0\}$ be a convex conic domain, $F
\mathcal{T\rightarrow }(0,\infty )$ a smooth positively 1-homogeneous
function such that the Hessian $g_{v}$ of $F^{2}$ has only one positive
eigenvalue for all $v\in \mathcal{T}$. Then, there hold:

(i) The fundamental (or reverse Cauchy-Schwarz) inequalit
\begin{equation}
dF_{v}(w)\geq F(w).  \label{reverse_CS_coord_free}
\end{equation}

(ii) The reverse triangle inequality
\begin{equation}
F(v+w)\geq F(v)+F(w);  \label{reverse triangle}
\end{equation
If the fundamental tensor $g_{v}$ is everywhere nondegenerate (i.e.,
Lorentzian), the above inequalities are strict.
\end{theorem}

\begin{proof}
\textit{(i)} The technique follows roughly the same steps as in the positive
definite case (see, e.g., \cite{Bao}, p. 8-9). Consider two arbitrary
vectors $u,v\in \mathcal{T}.$ Since $\mathcal{T}$ is convex, it follows that
$\dfrac{u+v}{2}\in \mathcal{T};$ but as it is also conic, we find $u+v\in
\mathcal{T},$ which means that it makes sense to speak about $F(u+v)$. Now,
perform a Taylor expansion around $v$, with the remainder in Lagrange form
\begin{equation}
F(u+v)=F(v)+F_{i}(v)u^{i}+\dfrac{1}{2}F_{ij}(v+\varepsilon u)u^{i}u^{j}.
\label{Taylor1}
\end{equation
From the above Lemma, we obtain that $F_{ij}$ is negative semidefinite, that
is, $F_{ij}(v+\varepsilon u)u^{i}u^{j}\leq 0$ and therefore
\begin{equation}
F(u+v)\leq F(v)+F_{i}(v)u^{i}.  \label{aux}
\end{equation}

Then, denoting $w:=u+v,$ the above becomes $F(w)\leq
F(v)+F_{i}(v)(w^{i}-v^{i}),$ which, using the 1-homogeneity of $F,$ leads
to: $F(w)\leq F_{i}(v)w^{i},$ which is the coordinate form of \textit{(i).}

\textit{(ii)} The reverse triangle inequality now follows similarly to the
nondegenerate case, see \cite{Minguzzi2014}. Choose any $v,w\in \mathcal{T}$
and set $\xi :=v+w.$ Since $\mathcal{T}$ is conic and convex, we have $\xi
\in \mathcal{T}$ and, by 1-homogeneity,
\begin{equation*}
F(\xi )=F_{i}(\xi )\xi ^{i}=F_{i}(\xi )(v^{i}+w^{i})=F_{i}(\xi
)v^{i}+F_{i}(\xi )w^{i}.
\end{equation*
Using the fundamental inequality twice in the right hand side, we get
\begin{equation*}
F(v+w)=F(\xi )\geq F(v)+F(w).
\end{equation*}

\textit{Strictness:\ }If $g_{v}$ is nondegenerate, then the equality
F_{ij}(v+\varepsilon u)u^{i}u^{j}=0$ can only happen when $v$ and $u$ are
collinear; in turn, this means that in (\ref{aux}), equality also happens
only if $u$ and $v$ are collinear. This leads to the strictness of (\re
{reverse_CS_coord_free}) and (\ref{reverse triangle}).
\end{proof}

\bigskip

Using (\ref{F_w}), the fundamental inequality can be equivalently written as
\begin{equation}
F_{i}(v)w^{i}\geq F(w),  \label{CS_coords}
\end{equation
or as
\begin{equation*}
g_{v}(v,w)\geq F(v)F(w).
\end{equation*}

\bigskip

\textbf{Example. }To convince ourselves that the reverse Cauchy-Schwarz
inequality becomes non-strict if $F\ $is degenerate, consider
\begin{equation*}
F:\mathcal{T}\rightarrow \mathbb{R},~~F(v)=\sqrt{\left( v^{0}\right)
^{2}-\left( v^{1}\right) ^{2}-....-\left( v^{k}\right) ^{2}};
\end{equation*
here, $n>3,~k\leq n-2$ and the cone $\mathcal{T}\subset \mathbb{R}^{n+1}$ is
the Cartesian product $\mathcal{T}=\mathcal{T}_{k}\times \mathbb{R}^{n-k},$
where $\mathcal{T}_{k}=\{u\in \mathbb{R}^{k+1}~|~\left( u^{0}\right)
^{2}-\left( u^{1}\right) ^{2}-...-\left( u^{k}\right) ^{2}>0,~u^{0}>0\}.$
Since $\mathcal{T}_{k}$ is a convex cone in $\mathbb{R}^{k+1}$, it follows
that $\mathcal{T}$ is also convex. The corresponding metric tensor is
g_{v}=diag(1,-1,...,-1,0,...,0),$ where the number of $-1$ entries is $k.$
Picking $v=(1,0,0,....0,1,0)$ and $w=(1,0,0,....,0,1),$ we get:
g_{v}(v,w)=1,$ $F(v)=1$ and $F(w)=1,$ which means that $g_{v}(v,w)=F(v)F(w),$
while, obviously, $v$ and $w$ are not collinear.

In the maximally degenerate case, when $g_{v}$ has everywhere signature
(+,0,0,...,0),$ the $(F_{ij}(v))$ is the zero matrix, hence, (\re
{reverse_CS_coord_free}) and (\ref{reverse triangle}) become equalities for
\textit{all }$v,w\in \mathcal{T}.$

\bigskip

Similarly, it holds:

\begin{proposition}
\label{positive semidefinite case}\textbf{(The positive semidefinite case):}
Let $\mathcal{T}\subset V\backslash \{0\}$ be a convex conic domain and $F
\mathcal{T\rightarrow }(0,\infty ),$ a smooth positively 1-homogeneous
function such that the Hessian $g_{v}$ of $F^{2}$ is positive semidefinite
for all $v\in \mathcal{T}$. Then, the Cauchy-Schwarz inequality (\ref{pos
def CS Finsler})\ and the triangle inequality (\ref{pos def triangle ineq
Finsler}) still hold, but they are generally non-strict.
\end{proposition}

The proof is identical to the one of Proposition \ref{prop_degenerate
Finsler case}, with the only difference that, in (\ref{Taylor1}), the matrix
$F_{ij}(v+\varepsilon u)$ is positive semidefinite, which leads to the
opposite inequality: $F(u+v)\geq F(v)+F_{i}(v)u^{i},$ hence, to the usual
(non-reversed)\ Cauchy-Schwarz and triangle inequalities.

\bigskip

Here are two refinements of the reverse triangle inequality, holding for
(possibly degenerate) Lorentz-Finsler functions.

\begin{theorem}
If the smooth, positively 1-homogeneous\textbf{\ }function $F:\mathcal{T
\rightarrow (0,\infty )$ defined on a convex conic domain $\mathcal{T\subset
}V\backslash \{0\}$ has the property that $g_{v}=\dfrac{1}{2}Hess(F^{2})$
has everywhere a single positive eigenvalue, then, for all $v,w\in \mathcal{
}$ and for any $0<a\leq b,$ we have
\begin{equation}
a\left[ F(v+w)-F(v)-F(w)\right] \leq F(av+bw)-aF(v)-bF(w)\leq
b[F(v+w)-F(v)-F(w)].  \label{reverse triangle_1}
\end{equation
Moreover, if $Hess(F^{2})$ is nondegenerate, i.e., $F$ is a Lorentz-Finsler
norm, then the inequalities are strict.
\end{theorem}

\begin{proof}
The first inequality is equivalent (after canceling out the $-aF(v)$ terms
and grouping the $F(w)$ ones into the left hand side) to:
\begin{equation*}
aF(v+w)+(b-a)F(w)\leq F(av+bw).
\end{equation*}

But, since $F$ is positively homogeneous and $b-a\geq 0,$ we get:
aF(v+w)=F(av+aw)$ and $(b-a)F(w)=F(bw-aw)$. Then, the reverse triangle
inequality yields:
\begin{equation*}
aF(v+w)+(b-a)F(w)=F(av+aw)+F(bw-aw)\leq F(av+bw)
\end{equation*
as required.

The second inequality is proven in a completely similar way to be equivalent
to: $F(av+bw)+F(bv-av)\leq F(bv+bw),$ which, again, holds by virtue of the
reverse triangle inequality.
\end{proof}

\begin{proposition}
With the assumptions from above Theorem, we have:
\begin{equation}
F(v)+F(w)\leq 2\int_{0}^{1}F(tv+(1-t)w)dt\leq F\left( v+w\right) ,
\label{reverse triangle_3}
\end{equation
for all $v,w\in \mathcal{T\subset }$ $V\backslash \{0\}$.
\end{proposition}

\begin{proof}
We use the same idea as in \cite{Min-Pal}. As we have seen above, the
Hessian $\left( F_{ij}\right) $ of $F$ is negative semidefinite, i.e., the
function $F$ is concave. Therefore:
\begin{equation*}
F(tv+(1-t)w))\geq tF(v)+(1-t)F(w),
\end{equation*
for every $v,w\in \mathcal{T}$, $t\in \lbrack 0,1]$. Integrating with
respect to $t,$ from $0$ to $1$, we obtain:
\begin{equation*}
\frac{F(v)+F(w)}{2}\leq \int_{0}^{1}F(tv+(1-t)w)dt,
\end{equation*
i.e., the first inequality (\ref{reverse triangle_3}). Further, using the
reverse triangle inequality, we have $F(v+w)=F{\large (}v+(1-t)w+(1-t)v+t
{\large )}\geq F(tv+(1-t)w)+F((1-t)v+tw)$. Integrating from $0$ to $1$, we
deduce: $F(v+w)\geq
\dint_{0}^{1}F(tv+(1-t)w)dt+\dint_{0}^{1}F((1-t)v+tw)dt=
\dint_{0}^{1}F(tv+(1-t)w)dt$, which is just the second inequality (\re
{reverse triangle_3}).
\end{proof}

The two above results trivially hold when one of the vectors $v,w$ is zero.

\section{Lorentz-Finsler norms and their inequalities}

\label{sec:ex}

The set of Lorentz-Finsler norms is rich in interesting examples whose
reverse Cauchy-Schwarz or triangle inequality yields immediately famous
inequalities from the literature and opens a pathway to reveal further
interesting inequalities in a simple way.

We already pointed out in Section \ref{sec:classineq} that, for the simplest
example of Lorentz-Finsler structure on $\mathbb{R}^{n+1}$, the Minkowski
metric $F(v)=\sqrt{\eta _{ij}v^{i}v^{j}}$, for which $\mathcal{T=}\left\{
v\in V~|~\eta _{ij}v^{i}v^{j}>0,v^{0}>0\right\} $, its reverse
Cauchy-Schwarz inequality led directly to Acz\'{e}l's inequality (\ref{1_1}).

In the following, we explore some nontrivial Finslerian cases.

\subsection{Popoviciu's inequality\label{Section_Popoviciu}}

\begin{proposition}
Let $\mathcal{T}\subset \mathbb{R}^{n+1}$ be the conic domain:
\begin{equation*}
\mathcal{T}:=\{v\in \mathbb{R}^{n+1}~|~v^{0},v^{1},...,v^{n}>0,\left(
v^{0}\right) ^{p}-\left( v^{1}\right) ^{p}-...-\left( v^{n}\right) ^{p}>0\}.
\end{equation*
Moreover let $F:\mathcal{T}\rightarrow \mathbb{R}^{+}$ be the Finsler
structure defined by
\begin{equation}
F(v)=H(v)^{\frac{1}{p}},\quad H(v)=\left( v^{0}\right) ^{p}-\left(
v^{1}\right) ^{p}-...-\left( v^{n}\right) ^{p}\,,  \label{p-pseudo-norm}
\end{equation
where $p>1$. Then:

(i) the fundamental inequality $F_{i}(v)w^{i}\geq F(w)$ is Popoviciu's
inequality:
\begin{equation}
\eta _{ij}a^{i}b^{j}\geq \left[ (a^{0})^{q}-\left( a^{1}\right)
^{q}-...-\left( a^{n}\right) ^{q}\right] ^{\frac{1}{q}}\left[ \left(
b^{0}\right) ^{p}-\left( b^{1}\right) ^{p}-...-\left( b^{n}\right) ^{p
\right] ^{\frac{1}{p}},\ \forall a,b\in \mathcal{T}\,,
\label{Popoviciu inequality}
\end{equation
where $\dfrac{1}{p}+\dfrac{1}{q}=1$;

(ii) the reverse triangle inequality of $F$ is Bellmann's inequality:
\begin{equation}
\left( v_{0}^{p}-v_{1}^{p}-...-v_{n}^{p}\right) ^{1/p}+\left(
w_{0}^{p}-w_{1}^{p}-...-w_{n}^{p}\right) ^{1/p}\leq \lbrack
(v_{0}+w_{0})^{p}-(v_{1}+w_{1})^{p}-...-(v_{n}+w_{n})^{p}]^{1/p}.
\label{Bellmann}
\end{equation}
\end{proposition}

\begin{proof}
\textit{(i) }To see that the fundamental inequality holds, we realise that
the Hessian of $H$ is\textbf{\
\begin{equation*}
H_{ij}(v)=p(p-1)diag\left(
(v^{0})^{p-2},-(v^{1})^{p-2},....,-(v^{n})^{p-2}\right) .
\end{equation*
and thus has Lorentzian signature on $\mathcal{T}$. Moreover, $T$ is convex
(but not strictly convex), as it can be identified with the epigraph of the
convex function $\tilde{H}(v^{1},...,v^{n})=(v^{1})^{p}+....+\left(
v^{n}\right) ^{p}$, defined for all $v^{\alpha }>0$.

By Proposition~\ref{prop:signHg} (see Appendix), we obtain that $g_{ij}(v)$
is Lorentzian for all $v\in \mathcal{T}$ and hence, the fundamental
inequality inequality (\ref{CS_coords}) holds.

By a straightforward calculation, we get:
\begin{equation*}
F_{i}(v)=F(v)^{1-p}\eta _{ij}(v^{j})^{p-1};
\end{equation*
therefore, the fundamental inequality $F_{i}(v)w^{i}\geq F(w)$ becomes:
\begin{equation*}
\eta _{ij}(v^{j})^{p-1}w^{i}\geq F(v)^{p-1}F(w)=H(v)^{\frac{p-1}{p}}H(w)^
\frac{1}{p}}.
\end{equation*
Evaluating the above equation with the following notation:
\begin{equation*}
q:=\dfrac{p}{p-1},\quad a^{i}:=\left( v^{i}\right) ^{p-1},\quad \
b^{j}:=w^{j},
\end{equation*
(in particular, $\dfrac{1}{p}+\dfrac{1}{q}=1$), yields Popoviciu's
inequality.

Statement \textit{(ii) }is obvious.
\end{proof}

\begin{remark}
Similarly, H\"{o}lder's inequality
\begin{equation*}
\delta _{ij}a^{i}b^{j}\leq \left[ \left( a^{0}\right) ^{q}+...+\left(
a^{n}\right) ^{q}\right] ^{\frac{1}{q}}\left[ \left( b^{0}\right)
^{p}+...+\left( b^{n}\right) ^{p}\right] ^{\frac{1}{p}},\ \ \ \ \ \forall
a^{i},b^{i}>0,i=\overline{0,n},
\end{equation*
where $\delta _{ij}$ is the Kronecker symbol, can be treated as fundamental
inequality of the Finsler norm $F(v)=[\left( v^{0}\right) ^{p}+...+\left(
v^{n}\right) ^{p}]^{\tfrac{1}{p}}$ - which is positive definite for $v^{i}>0,
$ $i=\overline{0,n}$ and Minkowski's inequalit
\begin{equation*}
\left[ \left( a^{0}+b^{0}\right) ^{p}+...+\left( a^{n}+b^{n}\right) ^{p
\right] ^{\frac{1}{p}}\leq \left[ \left( a^{0}\right) ^{p}+...+\left(
a^{n}\right) ^{p}\right] ^{\frac{1}{p}}+\left[ \left( b^{0}\right)
^{p}+...+\left( b^{n}\right) ^{p}\right] ^{\frac{1}{p}},
\end{equation*
$\forall a^{i},b^{i}>0,i=\overline{0,n},p>1$ is just the corresoponding
triangle inequality.
\end{remark}

\subsection{The aritmetic-geometric mean inequality\label{Section_BM}}

\begin{proposition}
Let $\mathcal{T}\subset \mathbb{R}^{n+1}$ be the convex conic domain
\begin{equation*}
\mathcal{T}:=\left\{ v\in \mathbb{R}^{n+1}~|~v^{0},v^{1},...,v^{n}>0\right\}
\subset \mathbb{R}^{n+1}\,.
\end{equation*
Moreover let $F:\mathcal{T}\rightarrow \mathbb{R}^{+}$ be the Berwald-Mo\'{o
r Finsler structure defined by
\begin{equation*}
F(v)=(v^{0}v^{1}...v^{n})^{\tfrac{1}{n+1}}\,.
\end{equation*
Then, the fundamental inequality $F_{i}(v)w^{i}\geq F(w)$ is the
aritmetic-geometric mean inequality:
\begin{equation}
\dfrac{a_{0}+....+a_{n}}{n+1}\geq \left( a_{0}a_{1}...a_{n}\right) ^{\tfrac{
}{n+1}},~\ \ \forall a_{i}\in \mathbb{R}_{+}^{\ast }\,.
\label{arithmetic-geometric mean}
\end{equation}
\end{proposition}

\begin{proof}
The $n$-dimensional Berwald-Mo\'{o}r metric is known, \cite{Asanov1980}, to
be of Lorentzian signature. Yet, for the sake of completeness, we sketch a
proof of this fact below. To this aim, we will use Proposition \re
{prop:signHg}\textbf{.}

The Hessian of the $(n+1)$-th power $H(v):=v^{0}v^{1}...v^{n+1}$ of $F$ is:
\begin{equation}
H_{ij}(v)=\left\{
\begin{array}{c}
0,~\ \ if~\ i=j \\
\dfrac{H(v)}{v^{i}v^{j}},\,\ \ \ \ if~~\ i\not=j
\end{array
\right.  \label{BM_H}
\end{equation
On $\mathcal{T},$ the matrix $(H_{ij}(v))$ has Lorentzian signature. To see
this, fix an arbitrary\textbf{\ }$v\in \mathcal{T}$ and introduce the
vectors $e_{0}:=v$ and$\ \ \{e_{\alpha }\},$ $\alpha =\overline{1,n}$ as
follows:
\begin{equation*}
e_{\alpha }^{i}=A_{\alpha }^{i}v^{i},~\ i=\overline{0,n}\text{ }
\end{equation*
(where no summation is understood over $i$), such that:
\begin{equation*}
\sum_{i=0}^{n}A_{\alpha }^{i}=A_{\alpha }^{0}+\sum_{\beta =1}^{n}A_{\alpha
}^{\beta }=0,~\ \ \ \ \ \det (A_{\alpha }^{\beta })_{\alpha ,\beta
\overline{1,n}}\not=0.
\end{equation*
The vectors $\left\{ e_{0},e_{\alpha }\right\} $ are linearly independent,
as the matrix with the columns $e_{0},e_{\alpha }$ has the determinant
H(v)\det (A_{\alpha }^{\beta })\not=0$. Moreover, $e_{\alpha }$ span the
(H_{ij}(v))$-orthogonal complement of $e_{0}=v$, since, using (\ref{BM_H}),
we find:
\begin{equation*}
H_{ij}(v)v^{i}e_{\alpha }^{j}=0,~\ \ \forall \alpha =1,...,n.
\end{equation*
Then, on one hand, we have
\begin{equation*}
H_{ij}(v)v^{i}v^{j}=n(n-1)H(v)>0
\end{equation*
and, on the other hand, $H_{ij}(v)$ is negative definite on $Span\{e_{\alpha
}\}$, since:
\begin{equation*}
H_{ij}(v)e_{\alpha }^{i}e_{\alpha }^{j}=H(v)\sum_{i\neq j}\frac{v^{i}}{v^{i}
\frac{v^{j}}{v^{j}}A_{\alpha }^{i}A_{\alpha }^{j}=H(v)\left( \left(
\sum_{i=0}^{n}A_{\alpha }^{i}\right) ^{2}-\sum_{i=0}^{n}(A_{\alpha
}^{i})^{2}\right) =0-H(v)\sum_{i=0}^{n}(A_{\alpha }^{i})^{2}\,.
\end{equation*
Consequently, $H_{ij}(v)$ has Lorentzian signature. Then, by Proposition \re
{prop:signHg}, also $g_{ij}(v)$ has Lorentzian signature on $\mathcal{T}$
and the fundamental inequality $F_{i}(v)w^{i}\geq F(w)$ holds $\forall
v,w\in \mathcal{T}$. We easily find:
\begin{equation*}
F_{i}(v)=\frac{1}{n+1}H^{\frac{1}{n+1}-1}\frac{H(v)}{v^{i}}=\frac{F(v)}{n+1
\frac{1}{v^{i}}
\end{equation*
and thus
\begin{equation*}
\frac{F(v)}{n+1}\sum_{i=0}^{n}\frac{w^{i}}{v^{i}}\geq F(w)
\end{equation*
or equivalently
\begin{equation*}
\frac{1}{n+1}\sum_{i=0}^{n}\frac{w^{i}}{v^{i}}\geq \frac{F(w)}{F(v)}=\left(
\dfrac{w^{0}}{v^{0}}\dfrac{w^{1}}{v^{1}}...\dfrac{w^{n}}{v^{n}}\right) ^
\tfrac{1}{n+1}}\,.
\end{equation*
Setting $a_{i}:=\dfrac{w^{i}}{v^{i}},\ i=\overline{0,n}$, $a_{i}$ take all
possible values in $\mathbb{R}_{+}^{\ast }$ and the fundamental inequality
becomes the aritmetic-geometric mean inequality
\eqref{arithmetic-geometric
mean}.
\end{proof}

\subsection{Weighted arithmetic-geometric mean inequality\labe
{Section_weighted}}

\begin{proposition}
Let $\mathcal{T}\subset \mathbb{R}^{n+1}$ be the convex conic domain
\begin{equation*}
\mathcal{T}:=\left\{ v\in \mathbb{R}^{n+1}~|~v^{0},v^{1},...,v^{n}>0\right\}
\subset \mathbb{R}^{n+1}\,.
\end{equation*
Then, the function $F:\mathcal{T}\rightarrow \mathbb{R}^{+},$ defined by
\begin{equation*}
F(v)=v^{a_{0}}v^{a_{1}}...v^{a_{n}},\ \sum_{i=0}^{n}a_{i}=1\quad a_{i}\geq
0\,.
\end{equation*
is a Lorentz-Finsler norm whose fundamental inequality is the weighted
arithmetic-geometric mean inequality
\begin{equation*}
\sum_{i=0}^{n}a_{i}v^{i}\geq
(v^{0})^{a_{0}}(v^{1})^{a_{1}}...(v^{n})^{a_{n}},~\ \ \ \ \ \ \ \ \ \ \
\forall v^{i}\in \mathbb{R}_{+}^{\ast }\,.
\end{equation*}
\end{proposition}

\begin{proof}
Fix $u\in \mathcal{T}.$ The components of the fundamental tensor are (no sum
convention is employed in the following expression)
\begin{equation*}
g_{ij}(u)=\frac{1}{2}\frac{\partial ^{2}F^{2}}{\partial u^{i}\partial u^{j}
=F(u)^{2}\left( \frac{2a_{i}a_{j}}{u^{i}u^{j}}-\frac{a_{i}\delta _{ij}}
(u^{i})^{2}}\right) \,.
\end{equation*
Its signature can be determined similarly to the previous case. Introducing
the vectors $e_{0}=u$ and $e_{\alpha },\alpha =\overline{1,n}$ with
components
\begin{equation*}
e_{\alpha }^{i}=B_{\alpha }^{i}u^{i}
\end{equation*
such that
\begin{equation}
\sum_{i=0}^{n}a_{i}B_{\alpha }^{i}=a_{0}B_{\alpha }^{0}+\sum_{\beta
=1}^{n}a_{\beta }B_{\alpha }^{\beta }=0,  \label{B}
\end{equation
and $\det (B_{\alpha }^{\beta })\not=0,$ the vectors $\left\{
e_{0},e_{\alpha }\right\} $ are linearly independent and
\begin{equation*}
g_{ij}(u)e_{0}^{i}e_{\alpha }^{j}=0\,.
\end{equation*
Thus, $\{e_{\alpha }\}_{\alpha =\overline{1,n}}$ span the orthogonal
complement of $e_{0}=u$. Moreover, $g_{ij}(u)$ is negative definite on
Span\{e_{\alpha }\}$, since, by (\ref{B}), we have:
\begin{equation*}
g_{ij}(u)e_{\alpha }^{i}e_{\alpha }^{j}=F(u)^{2}\left( 2\left(
\sum_{i=0}^{n}a_{i}B_{\alpha }^{i}\right) ^{2}-\sum_{i=0}^{n}a_{i}(B_{\alpha
}^{i})^{2}\right) =-F(u)^{2}\sum_{i=0}^{n}a_{i}(B_{\alpha }^{i})^{2}<0;
\end{equation*
taking into account that $g_{ij}(u)e_{0}^{i}e_{0}^{j}=F(u)^{2}>0,$ we obtain
that $g_{ij}(u)\ $is Lorentzian. Calculating
\begin{equation*}
F_{i}(u)=a_{i}\frac{F(u)}{u^{i}},
\end{equation*
the fundamental inequality \eqref{CS_coords} becomes
\begin{equation*}
F(u)\sum_{i=0}^{n}\frac{a_{i}w^{i}}{u^{i}}\geq F(w)\,.
\end{equation*
Introducing $v_{i}=\frac{w^{i}}{u^{i}}\in \mathbb{R}_{+}^{\ast },$ it can be
rewritten as: $\sum\limits_{i=0}^{n}a_{i}v^{i}\geq
(v^{0})^{a_{0}}(v^{1})^{a_{1}}...(v^{n})^{a_{n}}\,.$The generalization (\re
{weighted_a}) then follows immediately.
\end{proof}

\subsection{\label{Section_bimetric}Bimetric structures on $\mathbb{R}^{n+1}
}

On 4-dimensional spacetime manifolds, Finsler functions of the type
\begin{equation*}
F(v)=[(g_{ij}v^{i}v^{j})(h_{lk}v^{k}v^{l})]^{\frac{1}{4}}\,,
\end{equation*
where $g_{ij}$ and $h_{kl}$ are Lorentzian metrics of same signature type
(+,-,-,...,-)$, are relevant in physics when one describes the propagation
of light in birefringent crystals, see for example \cit
{PerlickBook,Pfeifer:2016har,Punzi:2007di}. We can always choose a basis the
tangent spaces of the manifold such that one of the bilinear forms $h$ or $g$
assumes its normal form, i.e. it is locally diagonal with entries
g_{ij}=\eta _{ij}$.

\begin{proposition}
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski metric $\eta $ and
another bilinear form $h$ of Lorentzian signature. Let $\mathcal{T}\subset
\mathbb{R}^{n+1}$ be the convex conic domain given by the intersection of
the future pointing causal vectors of $\eta $ and $h$
\begin{equation*}
\mathcal{T}:=\left\{ v\in \mathbb{R}^{n+1}|\eta _{ij}v^{i}v^{j}>0,\
h_{ij}v^{i}v^{j}>0,v^{0}>0\right\} \subset \mathbb{R}^{n+1}
\end{equation*
and $F:\mathcal{T}\rightarrow \mathbb{R}^{+},$ the bimetric Finsler
structure defined by
\begin{equation*}
F(v)=[(\eta _{ij}v^{i}v^{j})^{\frac{1}{4}}(h_{lk}v^{k}v^{l})]^{\frac{1}{4
}\,.
\end{equation*
Then, the fundamental inequality $F_{i}(v)w^{i}\geq F(w)$ is
\begin{equation}
\frac{1}{2}\left( \frac{\eta _{ij}w^{i}v^{j}}{\eta _{kl}v^{k}v^{l}}+\frac
h_{ij}w^{i}v^{j}}{h_{kl}v^{k}v^{l}}\right) \geq \frac{F(w)}{F(v)},~\ \ \ \ \
\ \forall v,w\in \mathcal{T}\,.
\end{equation}
\end{proposition}

\begin{proof}
The Finsler structure is built from a fourth order polynomial $H(v):=(\eta
_{ij}v^{i}v^{j})(h_{lk}v^{k}v^{l})$ whose Hessian is given by
\begin{equation*}
H_{ij}=2\eta _{ij}(h_{lk}v^{k}v^{l})+4(\eta _{ik}h_{jl}+\eta
_{jk}h_{il})v^{k}v^{l}+2h_{ij}(\eta _{kl}v^{k}v^{l});
\end{equation*
it was proven, see \cite{Pfeifer:2011tk,Pfeifer:2013gha}, that $H_{ij}$ has
Lorentzian signature on $\mathcal{T}.$ Consequently, again using Proposition
\ref{prop:signHg}, $g_{ij}$ is of Lorentzian signature on $\mathcal{T}$ and
the fundamental inequality $F_{i}(v)w^{i}\geq F(w)$ holds. We easily
calculate:
\begin{equation*}
F_{i}(v)=\frac{1}{2}\frac{1}{H(v)^{\frac{3}{4}}}\left( \eta _{ij}v^{j}\
(h_{lk}v^{k}v^{l})+(\eta _{kl}v^{k}v^{l})\ h_{ij}v^{j}\right) \,,
\end{equation*
and thus the inequality
\begin{equation*}
\frac{1}{2}\frac{1}{H(v)^{\frac{3}{4}}}\left( \eta _{ij}w^{i}v^{j}\
(h_{lk}v^{k}v^{l})+(\eta _{kl}v^{k}v^{l})\ h_{ij}w^{i}v^{j}\right) \geq
H(w)^{\frac{1}{4}}
\end{equation*
holds. Both sides of the inequality can be multiplied with the positive
factor $H(v)^{-\frac{1}{4}}$ to obtain
\begin{equation*}
\frac{1}{2}\left( \frac{\eta _{ij}w^{i}v^{j}}{\eta _{kl}v^{k}v^{l}}+\frac
h_{ij}w^{i}v^{j}}{h_{kl}v^{k}v^{l}}\right) \geq \left( \frac{H(w)}{H(v)
\right) ^{\frac{1}{4}}\,.
\end{equation*}
\end{proof}

\bigskip

As an explicit example, take $\mathbb{R}^{2}$ with $v=(v^{0},v^{1})$,
w=(w^{0},w^{1})$ and set $h_{kl}v^{k}v^{l}:=2(v^{0})^{2}-(v^{1})^{2}.$ As
\eta _{ij}v^{i}v^{j}=(v^{0})^{2}-(v^{1})^{2}$, we find for all
v^{0},v^{1},w^{0},w^{1}$ such that $(v^{0})^{2}>(v^{1})^{2}$ and
(w^{0})^{2}>(w^{1})^{2}:
\begin{equation}
\frac{1}{16}\left( \frac{v^{0}w^{0}-v^{1}w^{1}}{(v^{0})^{2}-(v^{1})^{2}}
\frac{2v^{0}w^{0}-v^{1}w^{1}}{2(v^{0})^{2}-(v^{1})^{2}}\right) ^{4}\geq
\frac{((w^{0})^{2}-(w^{1})^{2})(2(w^{0})^{2}-(w^{1})^{2})}
((v^{0})^{2}-(v^{1})^{2})(2(v^{0})^{2}-(v^{1})^{2})}\,.
\end{equation}

\subsection{\label{Section_Kropina}Kropina metric}

We will use a Kropina-type deformation of the Minkowski metric $\eta $ in
order to find out an inequality regarding $\eta .$

\begin{proposition}
Let $\mathcal{T}\subset \mathbb{R}^{n+1}$ be the convex conic domain
\begin{equation*}
\mathcal{T}:=\left\{ v\in \mathbb{R}^{n+1}~|~\eta
_{ij}v^{i}v^{j}>0,v^{0}>0\right\} \subset \mathbb{R}^{n+1}\,
\end{equation*
and the smooth, 1-homogeneous function $F:\mathcal{T}\rightarrow \mathbb{R
^{+}$ be defined by
\begin{equation*}
F(v)=\frac{\eta _{ij}v^{i}v^{j}}{v^{0}}=\dfrac{1}{v^{0}
[(v^{0})^{2}-(v^{1})^{2}-....-(v^{1})^{2}].
\end{equation*
Then, $F$ obeys the non-strict fundamental inequality (\re
{reverse_CS_coord_free}), which becomes
\begin{equation}
2\eta (v,w)\geq \frac{w^{0}}{v^{0}}\eta (v,v)+\dfrac{v^{0}}{w^{0}}\eta
(w,w),~\ \ \ \ \forall v,w\in \mathcal{T}.  \label{eq:kropineq}
\end{equation}
\end{proposition}

\begin{proof}
Let us rewrite $F$ as
\begin{equation*}
F(v)=v^{0}-\dfrac{\vec{v}\cdot \vec{v}}{v^{0}},
\end{equation*
where $v=(v^{0},\vec{v}),$ $\vec{v}=(v^{1},...,v^{n})$ and $\vec{v}\cdot
\vec{v}=\delta _{\alpha \beta }v^{\alpha }v^{\beta }$ denotes the standard
Euclidean product on $\mathbb{R}^{n}.$ We easily get the derivatives of $F$
as:
\begin{eqnarray}
F_{0}(v) &=&1+\dfrac{\vec{v}\cdot \vec{v}}{(v^{0})^{2}},~\ \ \ F_{\alpha
}(v)=-2\dfrac{\delta _{\alpha \beta }v^{\beta }}{v^{0}},~\ \ \alpha =1,...,n;
\label{derivs_F_1} \\
F_{00}(v) &=&-2\dfrac{\vec{v}\cdot \vec{v}}{(v^{0})^{3}},~\ \ F_{\alpha 0}=
\dfrac{\delta _{\alpha \beta }v^{\beta }}{(v^{0})^{2}},~\ \ \ F_{\alpha
\beta }=-2\dfrac{\delta _{\alpha \beta }}{v^{0}}.  \label{derivs_F_2}
\end{eqnarray
Now, we can prove that the matrix $g_{ij}(v)$ has Lorentzian signature for
all $v\in \mathcal{T}.$

First, we notice that $g_{v}(v,v)=g_{ij}(v)v^{i}v^{j}=F(v)^{2}>0$ on
\mathcal{T}$.

Second, we show that $g_{v}$ is negative semidefinite on the $g_{v}
-orthogonal complement of $v.$ This complement is defined by
g_{ij}(v)v^{j}w^{i}=0$, which is equivalent to: $F_{i}(v)w^{i}=0;$ taking
into account the identity $g_{ij}(v)=FF_{ij}(v)+F_{i}(v)F_{j}(v)$, for $g_{v}
$-orthogonal vectors $w,$ we can write
\begin{equation}
g_{ij}(v)w^{i}w^{j}=F(v)(F_{ij}(v)w^{i}w^{j}).  \label{g_rel_Kropina}
\end{equation
Taking into account (\ref{derivs_F_2}), we get:
\begin{eqnarray*}
F_{ij}(v)w^{i}w^{j} &=&\dfrac{-2}{(v^{0})^{3}}\left[ (\vec{v}\cdot \vec{v
)\left( w^{0}\right) ^{2}-2\left( \vec{w}\cdot \vec{v}\right)
v^{0}w^{0}+\left( \vec{w}\cdot \vec{w}\right) (v^{0})^{2}\right]  \\
&=&\dfrac{-2}{(v^{0})^{3}}\left[ w^{0}(\vec{v}\cdot \vec{v})-v^{0}\left(
\vec{w}\cdot \vec{w}\right) \right] ^{2}\leq 0.
\end{eqnarray*
Since $F(v)>0$ on $\mathcal{T},$ we obtain from (\ref{g_rel_Kropina}) that
g_{ij}(v)w^{i}w^{j}\leq 0$ on the $g_{v}$-orthogonal complement of $v$.

The fundamental inequality can quickly be calculated with help of
\begin{equation*}
F_{i}(v)=2\frac{\eta _{ij}v^{j}}{v^{0}}-\frac{\eta _{jk}v^{j}v^{k}}
(v^{0})^{2}}\delta _{i}^{0}
\end{equation*
which eventually leads to the desired inequality \eqref{eq:kropineq}.
\end{proof}

\section{\label{Aczel section}Acz\'{e}l inequality: generalizations and
refinements}

In the following, we will use a particular class of Lorentzian Finsler
metrics to obtain a generalization and some refinements of the Acz\'{e}l
inequality (\ref{1_1}).

Consider, on $\mathbb{R}^{n+1}\backslash \{0\},$ a smooth, positive definite
Finsler norm $\bar{F}$ and set
\begin{equation}
F(v):=\sqrt{\left( v^{0}\right) ^{2}-\bar{F}^{2}(\vec{v})},  \label{smooth F}
\end{equation
where $v=\left( v^{0},\vec{v}\right) $ belongs to the open conic subset of
\mathbb{R}^{n+1}\backslash \{0\}:
\begin{equation}
\mathcal{T}=\left\{ v=\left( v^{0},\vec{v}\right) \in \mathbb{R
^{n+1}~|v^{0}>\bar{F}(\vec{v})\right\} .  \label{T_set_Aczel}
\end{equation
Such functions $F$ belong to the class of stationary Lorentz Finsler norms
studied in \cite{Lammerzahl:2012kw,Stancarone}. As the function $\bar{F}$\
is convex, its epigraph $T$ is convex. Moreover, it is connected since it
can be characterized as the preimage of $(0,\infty )$ through the continuous
function $v\mapsto \left( v^{0}\right) -\bar{F}(\vec{v})$\textbf{. }Hence,
\mathcal{T}$ is a convex conic domain.

The metric tensor $g=\dfrac{1}{2}Hess(F^{2})$ is Lorentzian on $\mathcal{T},$
except for the half line $\left\{ \left( v^{0},0,0,....,0\right)
~|v^{0}>0\right\} .$ That is, it will obey on $\mathcal{T}$ the non-strict
version of the reverse Cauchy-Schwarz inequality. As $g_{00}=1,g_{0\alpha
}=0 $ and $g_{\alpha \beta }=-\dfrac{1}{2}\dfrac{\partial ^{2}\bar{F}^{2}}
\partial v^{\alpha }\partial v^{\beta }},$ we immediately obtain:

\begin{proposition}
\textbf{(Finslerian Acz\'{e}l inequality)}\textit{: Let }$\bar{F}:\mathbb{R
^{n}\rightarrow \mathbb{R}$ be an arbitrary Finsler norm on $\mathbb{R}^{n}$
and se
\begin{equation}
\left\Vert v\right\Vert =\bar{F}(\vec{v}),~\ \ \ \ \ \bar{g}_{\vec{v}}(\vec{
},\vec{w}):=\dfrac{1}{2}\dfrac{\partial ^{2}\bar{F}^{2}}{\partial v^{\alpha
}\partial v^{\beta }}(\vec{v})v^{\alpha }w^{\beta }.  \label{alpha_norm}
\end{equation
Then, for all $v^{0},w^{0}>0$ and for all $\vec{v},\vec{w}\in \mathbb{R}^{n}$
such that $\left( v^{0}\right) ^{2}-\left\Vert \vec{v}\right\Vert ^{2}>0,$
\left( w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2}>0$, there holds
\begin{equation}
\left[ v^{0}w^{0}-\vec{g}_{\vec{v}}(\vec{v},\vec{w})\right] ^{2}-[\left(
v^{0}\right) ^{2}-\left\Vert \vec{v}\right\Vert ^{2}][\left( w^{0}\right)
^{2}-\left\Vert \vec{w}\right\Vert ^{2}]\geq 0.  \label{Finslerian Aczel}
\end{equation}
\end{proposition}

\bigskip

In the following, we will obtain two refinements of the above inequality,
based on the following lemma:

\begin{lemma}
For all $v,w\in \mathbb{R}^{n+1},$ $v=(v^{0},\vec{v}),$ $w=(w^{0},\vec{w})$
with $\vec{w}\not=0,$ there holds
\begin{eqnarray}
&&\left[ v^{0}w^{0}-\vec{g}_{\vec{v}}(\vec{v},\vec{w})\right] ^{2}-[\left(
v^{0}\right) ^{2}-\left\Vert \vec{v}\right\Vert ^{2}][\left( w^{0}\right)
^{2}-\left\Vert \vec{w}\right\Vert ^{2}]=  \label{lemma} \\
&=&\left[ w^{0}\dfrac{\vec{g}_{\vec{v}}(\vec{v},\vec{w})}{\left\Vert \vec{w
\right\Vert }-v^{0}\left\Vert \vec{w}\right\Vert \right] ^{2}+\dfrac{\left(
w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2}}{\left\Vert \vec{w
\right\Vert ^{2}}\left[ \left\Vert \vec{v}\right\Vert ^{2}\left\Vert \vec{w
\right\Vert ^{2}-\vec{g}_{\vec{v}}(\vec{v},\vec{w})\right] .  \notag
\end{eqnarray}
\end{lemma}

\begin{proof}
By direct computation, we find that both hand sides are actually equal to:
\begin{equation}
\left[ \left\Vert \vec{v}\right\Vert ^{2}\left( w^{0}\right) ^{2}+\left(
v^{0}\right) ^{2}\left\Vert \vec{w}\right\Vert ^{2}-2v^{0}w^{0}\vec{g}_{\vec
v}}(\vec{v},\vec{w})\right] +\left[ \vec{g}_{\vec{v}}(\vec{v},\vec{w
)-\left\Vert \vec{v}\right\Vert ^{2}\left\Vert \vec{w}\right\Vert ^{2}\right]
.  \label{lhs_Aczel}
\end{equation}
\end{proof}

\begin{remark}
The Finslerian Acz\'{e}l inequality is actually strict. This can be seen as
follows. From the (non-reversed) Cauchy-Schwarz inequality for $\bar{F},$ we
find that both square brackets in (\ref{lhs_Aczel}) are positive. If both
these brackets are zero, then, $\vec{g}_{\vec{v}}(\vec{v},\vec{w
)-\left\Vert \vec{v}\right\Vert ^{2}\left\Vert \vec{w}\right\Vert ^{2}=0$
implies: $\vec{w}=\alpha \vec{v};$ the vanishing of the other bracket then
leads to $w^{0}=\alpha v^{0},$ i.e., the vectors $v,w\in \mathbb{R}^{n+1}$
are collinear.
\end{remark}

\bigskip

From the above lemma, we find:

\begin{proposition}
\textbf{(Refinements of the Finslerian Acz\'{e}l inequality)}. Let
\left\langle \cdot ,\cdot \right\rangle _{v}$ be defined by a Finslerian
norm $\bar{F}=\left\Vert \cdot \right\Vert $ on $\mathbb{R}^{n}$ as in (\re
{alpha_norm}). Then, for any $v^{0},w^{0}\in \mathbb{R}$ and any $\vec{v}
\vec{w}\in \mathbb{R}^{n}$, $\vec{w}\not=0,$\textbf{\ }such that\textbf{\ }
\left( v^{0}\right) ^{2}-\left\Vert \vec{v}\right\Vert ^{2}>0,$ $\left(
w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2}>0$\textbf{], }there
hold the inequalities:

(i) $\left[ v^{0}w^{0}-\vec{g}_{\vec{v}}(\vec{v},\vec{w})\right]
^{2}-[\left( v^{0}\right) ^{2}-\left\Vert \vec{v}\right\Vert ^{2}][\left(
w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2}]\geq \dfrac{\left(
w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2}}{\left\Vert \vec{w
\right\Vert ^{2}}(\left\Vert \vec{v}\right\Vert ^{2}\left\Vert \vec{w
\right\Vert ^{2}-\vec{g}_{\vec{v}}(\vec{v},\vec{w}));$

(ii) $\left[ v^{0}w^{0}-\vec{g}_{\vec{v}}(\vec{v},\vec{w})\right]
^{2}-[\left( v^{0}\right) ^{2}-\left\Vert \vec{v}\right\Vert ^{2}][\left(
w^{0}\right) ^{2}-\left\Vert \vec{w}\right\Vert ^{2}]\geq \lbrack w^{0
\dfrac{\vec{g}_{\vec{v}}(\vec{v},\vec{w})}{\left\Vert \vec{w}\right\Vert
-v^{0}\left\Vert \vec{w}\right\Vert ]^{2}.$
\end{proposition}

\bigskip

\textbf{Acknowledgements} C.P. was supported by the Estonian Ministry for
Education and Science through the Personal Research Funding Grants PSG489,
as well as the European Regional Development Fund through the Center of
Excellence TK133 \textquotedblleft The Dark Side of the
Universe\textquotedblright . N.V. was supported by a local grant of
Transilvania University\textbf{.}

\begin{appendices}
\section{The signature of $m$-th root metrics}

\label{app:mthroot} Among the Lorentz-Finsler norms in Section~\ref{sec:ex},
we encountered $m$-th root metrics, which are functions of the type:
\begin{equation*}
F:\mathcal{T}\rightarrow \mathbb{R},\quad F(v)=H(v)^{\tfrac{1}{m}},\quad
\text{with}\quad H(v):=a_{i_{1}...i_{m}}v^{i_{1}}...v^{i_{m}}\,,
\end{equation*
where $a_{i_{1}...i_{m}}$ are constants and $\mathcal{T}$ is a convex conic
domain in\textbf{\ }$\mathbb{R}^{n+1}$\textbf{\ }where\textbf{\ }$H(v)>0$.
Here, $m>2$ is fixed.

To determine the signature of $g_{ij}=\frac{1}{2}\frac{\partial F^{2}}
\partial v^{i}\partial v^{j}}$ in some of our examples we relate it to the
signature of the Hessian $H_{ij}:=\frac{\partial H}{\partial v^{i}\partial
v^{j}}$ of $H$. By direct calculation, we get (see also \cite{Pfeifer:2011tk
, \cite{Brinzei-projective}):
\begin{equation}
H_{ij}=mF^{m-2}[g_{ij}+(m-2)F_{i}F_{j}].  \label{P_ij}
\end{equation
The following result will greatly help us simplify calculations in concrete
examples.

\begin{proposition}
\label{prop:signHg} If, for a vector $v\in \mathcal{T},$ the matrix
H_{ij}(v)$ has Lorentzian signature $\left( +,-,-,..,-\right) ,$ then
g_{ij}(v)$ has also Lorentzian signature.
\end{proposition}

\begin{proof}
Assume that $H_{ij}(v)$ has index $n$; therefore, there exists an $n$
-dimensional subspace of $\mathbb{R}^{n+1}$ where it is negative definite.
Pick any vector $w\in \mathcal{T}$ with the property $H_{ij}(v)w^{i}w^{j}<0
. From (\ref{P_ij}) we find:
g_{ij}(v)w^{i}w^{j}+(m-2)(l_{i}(v)w^{j})^{2}<0. $ Since $m>2,$ this implies
g_{ij}(v)w^{i}w^{j}<0,$ that is, $(g_{ij})$ is also negative definite on at
least the same $n$-dimensional subspace.

But,\ $g_{ij}(v)v^{i}v^{j}=F^{2}(v)>0,$ which means that $g_{ij}$ cannot be
negative (semi-)definite on the whole $V.$ Consequently, it must be
Lorentzian.
\end{proof}
\end{appendices}


\section{Introduction}

Hadronic decays of $D$ mesons provide important information to understand the
weak and strong interactions in the charm sector. Various experiments have measured the branching fractions of
hadronic decays of $D$ mesons~\cite{pdg2014}. However, the measurement accuracy of the Cabibbo-favored (CF) decays $D\to \bar K\pi\eta^\prime$
is still very poor~\cite{pdg2014}. The Particle Data Group (PDG) gives a branching fraction of $(0.75\pm0.19)\%$ for $D^0\to K^-\pi^+\eta^\prime$,
which was measured by the CLEO collaboration 25 years ago~\cite{prd48_4007,pdg2014}. There are no measurements for the isospin-related decay modes
$D^0\to K^0_S\pi^0\eta^\prime$ and $D^+\to K^0_S\pi^+\eta^\prime$. The statistical isospin model (SIM) proposed in Refs.~\cite{rosner1,rosner2}
predicts a simple ratio of the branching fractions for the isospin multiplets: ${\mathcal B}(D^0\to K^-\pi^+\eta^\prime):{\mathcal B}(D^0\to K^0_S\pi^0\eta^\prime):{\mathcal B}(D^+\to K^0_S\pi^+\eta^\prime)
\equiv 1:{\mathcal R^0}:{\mathcal R^+}
\equiv 1:\frac{{\mathcal B}(D^0\to K^0_S\pi^0\eta^\prime)}{{\mathcal B}(D^0\to K^-\pi^+\eta^\prime)}:\frac{{\mathcal B}(D^+\to K^0_S\pi^+\eta^\prime)}{{\mathcal B}(D^0\to K^-\pi^+\eta^\prime)}=1:0.4:0.9$. Precision measurements of the branching fractions of $D\to \bar K\pi\eta^\prime$ are crucial to test the SIM prediction.

In this paper, we report an improved measurement of the branching fraction
for $D^0\to K^-\pi^+\eta^\prime$ and
the first measurements of the branching fractions for $D^0\to K^0_S\pi^0\eta^\prime$ and
$D^+\to K^0_S\pi^+\eta^\prime$.
The analysis is performed using an $e^+e^-$ annihilation data sample
corresponding to an integrated luminosity of $2.93$~fb$^{-1}$~\cite{lum}
collected with the BESIII detector~\cite{bes3} at $\sqrt s=3.773$~GeV.
At this energy,
relatively clean $D^0$ and $D^+$ meson samples are obtained
from the processes $e^+e^-\to\psi(3770)\to D^0\bar D^0$ or $D^+D^-$. To improve statistics,
we use a single-tag method, in which either a $D$ or $\bar D$ is reconstructed in an event.
Throughout the text, charge conjugated modes are implied, and
$D\bar D$ refers to $D^0\bar D^0$ and $D^+D^-$ unless stated explicitly.

\section{BESIII detector and Monte Carlo simulation}
\label{sec:detector}

The BESIII detector is a magnetic spectrometer
that operates at the BEPCII collider.  It has a cylindrical geometry with a solid-angle
coverage of 93\% of $4\pi$. It consists of several main components. A 43-layer
main drift chamber~(MDC) surrounding the beam pipe performs precise
determinations of charged particle trajectories and measures the
specific ionization energy loss~(${\rm d}E/{\rm d}x$) for charged particle identification~(PID).
An array of time-of-flight counters~(TOF) is located outside the MDC and provides additional PID
information. A CsI(Tl) electromagnetic calorimeter~(EMC) surrounds
the TOF and is used to measure the deposited energies of photons and
electrons. A solenoidal superconducting magnet outside the EMC
provides a 1 T magnetic field in the central tracking region of the
detector. The iron flux return of the magnet is instrumented with the resistive plate muon counters arranged in
nine layers in the barrel and eight layers in the endcaps for identification of
muons with momenta greater than 0.5\,GeV/$c$. More
details about the BESIII detector are described in Ref.~\cite{bes3}.

A Monte Carlo~(MC) simulation software package, based on {\sc geant4}~\cite{geant4},
includes the geometric description and response of the
detector and is used to determine the detection efficiency and
to estimate backgrounds for each decay mode.
An inclusive MC sample, which includes the $D^0\bar D^0$, $D^+D^-$ and non-$D\bar D$ decays of the
$\psi(3770)$, initial-state-radiation~(ISR) production of the
$\psi(3686)$ and $J/\psi$, the continuum process $e^+e^-\to q\bar q$~($q=u$,~$d$,~$s$), Bhabha scattering events, di-muon events and
di-tau events, is produced at $\sqrt s=3.773\,{\rm GeV}$.
The equivalent luminosity of the inclusive MC sample is ten times that of the data sample.
The $\psi(3770)$ decays are generated with the MC generator {\sc kkmc}~\cite{kkmc}, which
incorporates the effects of ISR~\cite{isr}.  Final-state-radiation\,(FSR)
effects are simulated with the \textsc{photos} package~\cite{photons}. The known decay modes are generated using
{\sc evtgen}~\cite{evtgen} with branching fractions taken from
the PDG~\cite{pdg2014}, while the remaining unknown decays are
generated using {\sc lundcharm}~\cite{lundcharm}.

\section{Event selection}
\label{sec:evtsel}
In this analysis, all charged tracks are required to be within
$|\rm{cos\theta}|<0.93$,
where $\theta$ is the polar angle with respect to the
positron beam.
Good charged tracks, except those used to reconstruct $K^0_{S}$
mesons, are required to originate from the interaction region defined by
$V_{xy}< 1$~cm and $|V_{z}|< 10$~cm,
where $V_{xy}$ and $|V_{z}|$ are the distances of the closest approach
of the reconstructed tracks to the interaction point (IP), perpendicular to
and along the beam direction, respectively.

Charged kaons and pions are
identified using the ${\rm d}E/{\rm d}x$ and TOF measurements.
The combined confidence levels
for the kaon and pion hypotheses ($CL_{K}$ and $CL_{\pi}$) are calculated and
the charged track is identified as kaon (pion) if $CL_{K(\pi)}$ is greater than
$CL_{\pi(K)}$.

The neutral kaon is reconstructed via the $K^0_S\to\pi^{+}\pi^{-}$ decay mode.
Two oppositely charged tracks with $|V_{z}|< 20$~cm are assumed to be a $\pi^+\pi^-$
pair without PID requirements
and the $\pi^+\pi^-$ pair is constrained to originate from a common vertex.
The $\pi^+\pi^-$ combination with an invariant mass $M_{\pi^+\pi^-}$ in the range
$|M_{\pi^+\pi^-}-M_{K^0_S}|<0.012$\,GeV/$c^2$, where $M_{K^0_S}$ is the nominal $K^0_{S}$ mass~\cite{pdg2014},
and
a measured flight distance from the IP greater than twice its resolution
is accepted as a $K^0_S$ candidate.
Figure~\ref{fig:sig}(a) shows the $\pi^+\pi^-$ invariant mass distribution, where the two solid arrows denote the $K^0_S$
signal region.

Photon candidates are selected using the EMC information.
The time of the candidate shower must be within 700\,ns of the event start time
and the shower energy should be greater than 25\,(50)\,MeV
if the crystal with the maximum deposited energy for the cluster of interest
is in the barrel~(endcap) region~\cite{bes3}.
The opening angle between the candidate shower and
any charged track is required to be greater than $10^{\circ}$
to eliminate showers associated with charged tracks.
Both $\pi^0$ and $\eta$ mesons are reconstructed via the $\gamma\gamma$ decay mode.
The $\gamma\gamma$ combination with an invariant mass within $(0.115,\,0.150)$
or $(0.515,\,0.570)$\,GeV$/c^{2}$ is regarded as a $\pi^0$ or $\eta$ candidate, respectively.
To improve resolution, a one constraint (1-C) kinematic fit is applied
to constrain the invariant mass of the photon pair to
the nominal $\pi^{0}$ or $\eta$ invariant mass~\cite{pdg2014}.

The $\eta^\prime$ mesons are reconstructed through the decay $\eta^\prime\to\pi^{+}\pi^{-}\eta$.
The invariant mass of the $\pi^{+}\pi^{-}\eta$ combination $M_{\pi^{+}\pi^{-}\eta}$
is required to satisfy $|M_{\pi^+\pi^-\eta}-M_{\eta^\prime}|<0.015$\,GeV/$c^2$,
where $M_{\eta^\prime}$ is the nominal $\eta^\prime$ mass~\cite{pdg2014}.
The boundaries of the one dimensional (1D) $\eta'$ signal region are illustrated by the two solid arrows shown in Fig.~\ref{fig:sig}(b).
The $D^{0(+)}\to K^-(K^0_S)\pi^+\eta^\prime$ decay is selected from the $K^-(K^0_S)\pi^+\pi^+\pi^-\eta$ combination.
Since the two $\pi^+$s in the event have low momenta and are indistinguishable,
the $\eta^\prime$ may be formed from either of the $\pi^+\pi^-\eta$ combinations, whose invariant masses
are denoted as $M_{\pi_1^{+}\pi^{-}\eta}$ and $M_{\pi_2^{+}\pi^{-}\eta}$.
Figure~\ref{fig:sig}(c) shows the scatter plot of
$M_{\pi_2^{+}\pi^{-}\eta}$ versus $M_{\pi_1^{+}\pi^{-}\eta}$
for the $D^0\to K^-\pi^+\eta^\prime$ candidate events in the data sample.
Events with at least one $\pi^+\pi^-\eta$ combination in the two dimensional (2D) $\eta^\prime$ signal region,
shown by the solid lines in Fig.~\ref{fig:sig}(c), are kept for further analysis.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=3.3in]{eta_ks_3_1.eps}
  \caption{ (Color online)\
(a)~Distribution of $M_{\pi^+\pi^-}$ for the $K^0_S$ candidates from $D^0\to K^0_S\pi^0\eta^\prime$ decays
and (b)~the combined $M_{\pi_{1}^+\pi^-\eta}$ and $M_{\pi_{2}^+\pi^-\eta}$ distribution for the $\eta^\prime$ candidates from $D^0\to K^-\pi^+\eta^\prime$
decays,
where the dots with error bars are data, the histograms are inclusive MC samples,
and the pairs of red solid~(blue dashed) arrows show the
boundaries of the $K^0_S$ or $\eta^\prime$ 1D signal~(sideband) region.
(c)~Scatter plot of $M_{\pi^+_2\pi^-\eta}$ versus $M_{\pi^+_1\pi^-\eta}$ for the $D^0\to K^-\pi^+\eta^\prime$  candidate events in the data sample, where
the range surrounded by the red solid~(blue dashed) lines denotes the $\eta^\prime$ 2D signal~(sideband) region.
In these figures,
except for the $K^0_S$ or $\eta^\prime$ mass requirement,
all selection criteria and an additional requirement of $|M_{\rm BC}-M_D|<0.005$~GeV/$c^2$ have been imposed.
The signal and sideband regions, illustrated here, are applied for all decays of interest in the analysis.
}\label{fig:sig}
\end{figure}

To distinguish $D$ mesons from backgrounds, we define two kinematic variables,
the energy difference $\Delta E \equiv E_D-E_{\rm beam}$ and
the beam-constrained mass
$M_{\rm BC} \equiv \sqrt{E^{2}_{\rm beam}-|\vec{p}_{D}|^{2}}$,
where $E_D$ and $\vec{p}_{D}$ are the energy and momentum
of the $D$ candidate in the $e^+e^-$ center-of-mass system and $E_{\rm beam}$ is the beam energy.
For each signal decay mode, only the combination with
the minimum $|\Delta E|$ is kept if more than one candidate passes
the selection requirements.
Mode-dependent $\Delta E$ requirements, as listed in Table~\ref{tab:singletagN_MC},
are applied to suppress combinatorial backgrounds.
These requirements are about $\pm 3.5\sigma_{\Delta E}$ around the fitted $\Delta E$ peaks, where $\sigma_{\Delta E}$ is the resolution of the $\Delta E$
distribution obtained from fits to the data sample.

\section{Data analysis} \label{sec:ana}
\begin{figure}[htbp]
  \centering
  \includegraphics[width=3.3in]{mbc_3_1.eps}
\caption{(Color online) Fits to the $M_{\rm BC}$ distributions of the (a)
$D^0\to K^-\pi^+\eta^\prime$, (b) $D^0\to K^0_S\pi^0\eta^\prime$, and
(c) $D^+\to K^0_S\pi^+\eta^\prime$ candidate events.
The dots with error bars are data,
the blue solid curves are the total fits
and the red dashed curves are the fitted backgrounds.
The dotted, dashed and solid histograms are
the scaled BKGI, BKGII, and BKGIII components~(see the last paragraph of Sec.~\ref{sec:evtsel}),
respectively.}\label{fig:datafit_Massbc}
\end{figure}

\begin{figure*}[htbp]
\centering
\includegraphics[width=7in]{mm_compare.eps}
\caption{
(Color online)
The $M_{K\pi}$,
$M_{\pi\eta^\prime}$, and $M_{K\eta^\prime}$ distributions of data (dots with error bars)
and MC simulations (histograms).
The top, middle, and bottom rows correspond to
$D^0\to K^-\pi^+\eta^\prime$, $D^0\to K^0_S\pi^0\eta^\prime$, and
$D^+\to K^0_S\pi^+\eta^\prime$ candidate events, respectively.
The blue dashed histograms are PHSP MC samples.
The red solid histograms are the modified MC samples.
The yellow shaded histograms are the backgrounds estimated from the inclusive MC sample.
An additional requirement of
$|M_{\rm BC}-M_D|<0.005$~GeV/$c^2$ has been imposed on the events shown in these plots.}\label{fig:mm_compare}
\end{figure*}

The $M_{\rm BC}$ distributions of the accepted candidate events for
the decays of interest in the data sample are shown in Fig.~\ref{fig:datafit_Massbc}.
Unbinned maximum likelihood fits to these spectra are performed
to obtain the $D$ signal yields.
In the fits, the $D$ signal is modeled by an MC-simulated shape convolved with a Gaussian function
with free parameters accounting for the difference between the detector resolution
of the data and that of the MC simulation.
The background shape is described by an ARGUS function~\cite{ARGUS}.
The potential peaking backgrounds are investigated as follows.
The combinatorial $\pi^+\pi^-$~(called BKGI) or $\pi^+\pi^-\eta$~(called BKGII)
pairs in the $K^0_S$ or $\eta^\prime$ signal region may survive
the event selection criteria and form peaking backgrounds around
the $D$ mass in the $M_{\rm BC}$ distributions.
These background components are validated by
the data events in the $K^0_S$($\eta^\prime$) sideband region
defined as $0.020\,(0.022)<|M_{\pi^+\pi^-\,(\pi^+\pi^-\eta)}-M_{K^0_S\,(\eta^\prime)}|<0.044\,(0.046)$\,GeV/$c^2$,
as indicated by the ranges between the adjacent pair of blue dashed arrows
in Fig.~\ref{fig:sig}(a)[(b)].
For $D^0\to K^-\pi^+\eta^\prime$ and $D^+\to K^0_S\pi^+\eta^\prime$ decays,
the data events in the $\eta^\prime$ 2D sideband region,
enclosed by the blue dashed lines in Fig.~\ref{fig:sig}(c), are examined.
For these events, either $M_{\pi_1^{+}\pi^{-}\eta}$ or $M_{\pi_2^{+}\pi^{-}\eta}$
is in the $\eta^\prime$ 1D sideband region, but
both are outside the $\eta^\prime$ 1D signal region.
These two background components are normalized by the ratios of the magnitude of the backgrounds
in the $K^0_S$\,($\eta^\prime$) signal and sideband regions.
The background components from other processes~(called BKGIII) are estimated by analyzing the inclusive MC sample.
The scaled $M_{\rm BC}$ distributions of the surviving events for the
BKGI, BKGII and BKGIII components are shown as
the dotted, dashed and solid histograms in Fig.~\ref{fig:datafit_Massbc}, respectively.
In these spectra, no peaking backgrounds are found,
which indicates that the background shape is well modeled by the ARGUS function.
From each fit, we obtain the number of $D\to \bar K\pi\eta^\prime$ signal events $N_{\rm tag}$,
as summarized in Table~\ref{tab:singletagN_MC}.
The statistical significances of these decays,
which are estimated from
the likelihood difference between the fits with and without
the signal component, are all greater than $10\sigma$.

Figure~\ref{fig:mm_compare} shows the $M_{K\pi}$,
$M_{\pi\eta^\prime}$, and $M_{K\eta^\prime}$ distributions
of $D\to \bar K\pi\eta^\prime$ candidate events for data and MC simulations after requiring $|M_{\rm BC}-M_D|<0.005$~GeV/$c^2$.
No obvious sub-resonances have been observed in these invariant mass distributions.
Nevertheless, the phase space (PHSP) MC distributions are not in good agreement with the data distribution (see the blue dashed histograms and dots with errors in Fig.~\ref{fig:mm_compare}).
To solve this problem, we modify the MC generator to produce the correct invariant mass distributions according to the Dalitz plot distributions in data.
In the Dalitz plot, the background component is modeled by the inclusive MC simulation, while the signal component is generated according to efficiency-corrected PHSP MC simulation. In Fig.~\ref{fig:dalitz}, we show the Dalitz plots of $D^0\to K^-\pi^+\eta^\prime$ candidate events for data and the modified MC sample. The invariant mass distributions $M_{K\pi}$, $M_{\pi\eta^\prime}$, and $M_{K\eta^\prime}$ of the modified MC samples are in good agreement with the data distributions (see the red solid histograms and dots with errors in Fig.~\ref{fig:mm_compare}). In the following, we use the modified MC sample to determine the detection efficiencies in the calculation of the branching fractions.

\begin{figure}[htbp]
\centering
\includegraphics[width=3.4in]{dalitz_in_mc2.eps}
\caption{\small
Dalitz plots of $M^2_{K^-\pi^+}$ vs. $M^2_{\pi^+\eta^\prime}$
for $D^0\to K^-\pi^+\eta^\prime$ candidate events in data (left) and modified MC sample (right). }\label{fig:dalitz}
\end{figure}

\section{Branching fractions}

The branching fraction of $D\to \bar K\pi\eta^\prime$ is determined according to
 \begin{equation}
 \label{equ:branchingfraction}
 {\mathcal B}(D\to \bar K\pi\eta^\prime) = \frac{N_{\rm tag{\color{blue}}}}{2\cdot N_{D\bar D}\cdot\epsilon
 \cdot{\mathcal B_{\eta^\prime}}
 \cdot{\mathcal B_{\eta}}
 (\cdot{\mathcal B_{\rm inter}})
},
 \end{equation}
where $N_{\rm tag}$ is the number of $D\to \bar K\pi\eta^\prime$ signal events,
$N_{D\bar D}$ is the total number of $D\bar D$ pairs,
$\epsilon$ is the detection efficiency which has been corrected by the differences in the
efficiencies for charged particle tracking and PID, as well as $\pi^0$ and $\eta$ reconstruction, between the data and MC simulation as discussed in Sec.~\ref{sec:ana}, and summarized in Table~\ref{tab:singletagN_MC}.
In Eq.~(\ref{equ:branchingfraction}), ${\mathcal B_{\rm inter}}$ is the product branching fraction $\mathcal B_{K^0_S}\cdot{\mathcal B_{\pi^0}}$\,($\mathcal B_{K^0_S}$)
for the decay $D^0\to K^0_S\pi^0\eta^\prime$\,($D^+\to K^0_S\pi^+\eta^\prime$), and
${\mathcal B_{\eta^\prime}}$, ${\mathcal B_{\eta}}$, ${\mathcal B_{K^0_S}}$
and ${\mathcal B_{\pi^0}}$ denote the branching fractions of the decays
$\eta^\prime\to\pi^+\pi^-\eta$,
$\eta\to\gamma\gamma$, $K^0_S \to \pi^+\pi^-$, and
$\pi^0\to\gamma\gamma$, respectively, taken from the PDG~\cite{pdg2014}.
With the single-tag method, the CF decays $D^0(D^+)\to \bar K\pi\eta^\prime$ are indistinguishable
from the doubly Cabibbo-suppressed (DCS) decays $\bar D^0(D^+)\to \bar K(K)\pi\eta^\prime$.
However, the DCS contributions are expected to be small and negligible in the calculations of branching fractions,
but will be taken into account as a systematic uncertainty.

Taking $N_{D^{0}\bar{D}^{0}}=(10597\pm28_{\rm stat.}\pm 98_{\rm syst.})\times 10^3$
and
$N_{D^{+}D^{-}}=(8296\pm31_{\rm stat.}\pm65_{\rm syst.})\times 10^3$ from Ref.~\cite{bes3ddyield},
the branching fraction of each decay is determined with Eq.~(\ref{equ:branchingfraction})
and summarized in Table~\ref{tab:singletagN_MC}.

\begin{table*}[htp]
\centering
\caption{\label{tab:singletagN_MC}
$\Delta E$ requirements, input quantities and results for the determination of the branching fractions.
The efficiencies do not include the branching fractions for the decays of the daughter particles of
$\eta^\prime$, $\eta$, $K^0_S$, and $\pi^0$ mesons.
The uncertainties are statistical only.
}
\small
\begin{tabular}{lcccc} \hline
\multicolumn{1}{c}{Decay mode}& $\Delta E$ (MeV)  & $N_{\rm tag}$ &$\epsilon$ (\%) &$\mathcal B$ ($\times 10^{-3}$)
\\ \hline
$D^0\to K^-\pi^+\eta^\prime$   &$(-26,+28)$ &$2528\pm59$  &$10.97\pm0.08$&$6.43\pm0.15$\\
$D^0\to K^0_S\pi^0\eta^\prime$ &$(-35,+38)$ &$289 \pm26$  &$4.67\pm0.04$&$2.52\pm0.22$\\
$D^+\to K^0_S\pi^+\eta^\prime$ &$(-27,+28)$ &$267 \pm24$  &$7.23\pm0.05$&$1.90\pm0.17$\\ \hline
\end{tabular}
\end{table*}

\section{Systematic uncertainties}
\label{sec:sys}
The systematic uncertainties in the measurements of the branching fractions and the branching ratios, ${\mathcal R}^0\equiv\frac{{\mathcal B}(D^0\to K^0_S\pi^0\eta^\prime)}{{\mathcal B}(D^0\to K^-\pi^+\eta^\prime)}$, and ${\mathcal R}^+\equiv \frac{{\mathcal B}(D^+\to K^0_S\pi^+\eta^\prime)}{{\mathcal B}(D^0\to K^-\pi^+\eta^\prime)} $, are summarized in Table~\ref{tab:relsysuncertainties}. Each contribution, estimated relative to the measured branching fraction, is discussed below.

\begin{table*}[htp]
\centering
\caption{\label{tab:relsysuncertainties} Relative systematic uncertainties (in \%) in the branching fractions, ${\mathcal R}^0$, and ${\mathcal R}^+$.
The numbers before or after `/' in the last two columns denote the remaining systematic uncertainties of
${\mathcal B}(D^0\to K^{-}\pi^{+}\eta')$ and ${\mathcal B}(D^{0(+)}\to K^{0}_{S}\pi^{0(+)}\eta')$ that do not cancel
in the determination of ${\mathcal R}^{0}$ and ${\mathcal R}^{+}$.     }
\begin{small}
\begin{tabular}{lccccc}
\hline
Source & ${\mathcal B}(D^0\to K^{-}\pi^{+}\eta')$&${\mathcal B}(D^0\to K^{0}_{S}\pi^{0}\eta')$&${\mathcal B}(D^+\to K^{0}_{S}\pi^{+}\eta')$&${\mathcal
R}^0$&${\mathcal R}^+$\\
\hline
  Number of $D\bar D$ pairs
                                 & 1.0  & 1.0 & 0.9 & --/-- &1.0/0.9 \\
  Tracking of $K^\pm(\pi^\pm)$   & 3.0  & 2.0 & 2.5 &1.0/-- &1.0/--  \\
  PID of $K^\pm(\pi^\pm)$        & 2.0  & 1.0 & 1.5 &1.0/-- &0.5/--  \\
  $K_S^0$ reconstruction         &  --  & 1.5 & 1.5 & --/1.5& --/1.5 \\
  $\pi^0\,(\eta)$ reconstruction & 1.0  & 2.0 & 1.0 & --/1.0& --/--  \\
  $M_{\rm BC}$ fit               & 0.5  & 3.6 & 1.9 &0.5/3.6&0.5/1.9 \\
  $\eta^\prime$ mass window      & 1.0  & 1.0 & 1.0 & --/-- & --/--  \\
  $\Delta E$ requirement         & 0.1  & 2.4 & 4.5 &0.1/2.4&0.1/4.5 \\
  MC modeling                    & 1.6  & 0.5 & 1.7 &1.6/0.5&1.6/1.7 \\
  MC statistics                  & 0.7  & 0.9 & 0.7 &0.7/0.9&0.7/0.7 \\
  Quoted branching fractions     & 1.7  & 1.7 & 1.7 & --/0.1& --/0.1 \\
  $D^0\bar D^0$ mixing           & 0.1  & 0.1 & --  & --/-- & --/--  \\
  DCS contribution& 0.6  & 0.6 & 0.6 & --/-- & --/--  \\
  \hline
  Total & 4.8 & 6.0 & 6.6 & 5.3 & 6.0 \\
  \hline
\end{tabular}
\end{small}
\end{table*}

\begin{itemize}

\item
{\bf \boldmath Number of $D\bar D$ pairs}:
The total numbers of $D^0\bar D^0$ and $D^+D^-$ pairs produced in
the data sample are cited from a previous measurement~\cite{bes3ddyield}
that uses a combined analysis of
both single-tag and double-tag events in the same data sample.
The total uncertainty in the quoted number of $D^0\bar D^0~(D^+D^-)$ pairs
is 1.0\%~(0.9\%), obtained by adding both the statistical and systematic uncertainties in quadrature.

\item
{\bf \boldmath Tracking and PID of $K^\pm(\pi^\pm)$}:
The tracking and PID efficiencies for $K^\pm(\pi^\pm)$
are investigated using double-tag $D\bar D$ hadronic events.
A small difference between the efficiency
in the data sample and that in MC simulation~(called the data-MC difference) is found.
The momentum weighted data-MC differences
in the tracking [PID] efficiencies are determined to be
$(+2.4\pm0.4)\%$, $(+1.0\pm0.5)\%$, and $(+1.9\pm1.0)\%$
[$(-0.2\pm0.1)\%$, $(-0.1\pm0.1)\%$ and $(-0.2\pm0.1)\%$]
for $K^\pm$, $\pi^\pm_{\rm direct}$, and $\pi^\pm_{\rm in-direct}$,
respectively.
Here, the uncertainties are statistical
and the subscript $_{\rm direct}$ or $_{\rm in-direct}$ indicates
the $\pi^\pm$ produced in $D$ or $\eta^\prime$ decays, respectively.
In this work, the MC efficiencies have been corrected by the
momentum weighted data--MC differences in the $K^\pm(\pi^\pm)$ tracking and PID efficiencies.
Finally, a systematic uncertainty for charged particle tracking is assigned to be 1.0\%
per $\pi^\pm_{\rm in-direct}$ and 0.5\% per $K^\pm$ or $\pi^\pm_{\rm direct}$.
The systematic uncertainty for PID efficiency is taken as 0.5\% per $K^\pm$, $\pi^\pm_{\rm direct}$ or $\pi^\pm_{\rm in-direct}$.

\item
{\bf\boldmath $K_S^0$ reconstruction}:
The $K_{S}^{0}$ reconstruction efficiency,
which includes effects from the track reconstruction of the charged pion pair, vertex fit, decay length requirement and $K^0_S$ mass window,
has been studied with a control sample of
$J/\psi\to K^{*}(892)^{\mp}K^{\pm}$ and $J/\psi\to \phi K_S^{0}K^{\pm}\pi^{\mp}$~\cite{sysks}.
The associated systematic uncertainty is assigned as 1.5\% per $K^0_S$.

\item
{\bf \boldmath $\pi^0\,(\eta)$ reconstruction}:
The $\pi^0$ reconstruction efficiency,
which includes effects from the photon selection, 1-C kinematic fit and $\pi^0$ mass window,
is verified with double-tag $D\bar D$ hadronic decay samples
of $D^0\to K^-\pi^+$, $K^-\pi^+\pi^+\pi^-$ versus
$\bar D^0\to K^+\pi^-\pi^0$, $K^0_S\pi^0$~\cite{syspi0}.
A small data-MC difference in the $\pi^0$ reconstruction efficiency
is found.
The momentum weighted data-MC difference in $\pi^0$ reconstruction
efficiencies is found to be $(-0.5\pm1.0)\%$, where the uncertainty
is statistical.
After correcting the MC efficiencies by the momentum weighted data-MC
difference in $\pi^0$ reconstruction efficiency,
the systematic uncertainty due to $\pi^0$ reconstruction is assigned as 1.0\% per $\pi^0$.
The systematic uncertainty due to $\eta$ reconstruction is assumed to be the same as that for $\pi^0$ reconstruction.

\item
{\bf \boldmath $\eta^\prime$ mass window}:
The uncertainty due to the $\eta^\prime$ mass window is studied by fitting to the $\pi^+\pi^-\eta$ invariant mass spectrum of the $K^-\pi^+\eta^\prime$
candidates. The difference between the data and MC simulation in the efficiency of the $\eta^\prime$ mass window restriction is $(0.8\pm0.2)$\%.
The associated systematic uncertainty is assigned as 1.0\%.

\item
{\bf \boldmath $M_{\rm BC}$ fit}:
To estimate the systematic uncertainty due to the $M_{\rm BC}$ fit, we repeat
the measurements by varying the fit range [$(1.8415,1.8865)$\,GeV/$c^2$],
the signal shape\,(with different MC matching requirements)
and the endpoint\,(1.8865\,GeV/$c^2$) of the ARGUS function ($\pm0.2$\,MeV/$c^2$).
Summing the relative changes in the branching fractions for these three sources in quadrature
yields 0.5\%, 3.6\%, and 1.9\% for
$D^0\to K^-\pi^+\eta^\prime$, $D^0\to K^0_S\pi^0\eta^\prime$, and
$D^+\to K^0_S\pi^+\eta^\prime$, respectively,
which are assigned as systematic uncertainties.

\item
{\bf \boldmath $\Delta E$ requirement}:
To investigate the systematic uncertainty due to the $\Delta E$ requirement,
we repeat the measurements with alternative $\Delta E$
requirements of $3.0\sigma_{\Delta E}$ and $4.0\sigma_{\Delta E}$ around the fitted $\Delta E$ peaks.
The changes in the branching fractions, 0.1\%, 2.4\%, and 4.5\%, are taken as systematic uncertainties for
$D^0\to K^-\pi^+\eta^\prime$, $D^0\to K^0_S\pi^0\eta^\prime$, and
$D^+\to K^0_S\pi^+\eta^\prime$, respectively.

\item
{\bf MC modeling}:
The systematic uncertainty in the MC modeling is studied by varying
MC-simulated background sizes for the input
$M^{2}_{K\pi}$ and $M^{2}_{\pi\eta^\prime}$ distributions in the generator by~$\pm20\%$.
The largest changes in the detection efficiencies, 1.6\%, 0.5\%, and 1.7\% are taken as systematic uncertainties
for $D^0\to K^-\pi^+\eta^\prime$, $D^0\to K^0_S\pi^0\eta^\prime$, and
$D^+\to K^0_S\pi^+\eta^\prime$, respectively.

\item
{\bf MC statistics}:
The uncertainties due to the limited MC statistics are 0.7\%, 0.9\% and 0.7\%
for $D^0\to K^-\pi^+\eta^\prime$, $D^0\to K^0_S\pi^0\eta^\prime$, and
$D^+\to K^0_S\pi^+\eta^\prime$, respectively.

\item
{\bf Quoted branching fractions}:
The uncertainties of the quoted branching fractions for
$\eta^\prime \to \pi^+\pi^-\eta$,
$\eta\to \gamma\gamma$,
$K^0_S\to \pi^+\pi^-$, and
$\pi^0\to \gamma\gamma$ are taken from the world average and are 1.6\%,
0.5\%, 0.07\%, and 0.03\%~\cite{pdg2014},
respectively.

\item
{\bf $D^0\bar D^0$ mixing}:
Because $D^0\bar D^0$ meson pair is coherently produced in $\psi(3770)$ decay,
the effect of $D^0\bar D^0$ mixing on the branching fractions of neutral $D$
meson decays is expected to be due to the
next-to-leading-order of the $D^0\bar D^0$ mixing parameters $x$ and $y$~\cite{zzxing,asner}.
With $x=(0.32\pm0.14)\%$ and $y=(0.69^{+0.06}_{-0.07})\%$ from PDG~\cite{pdg2014}, we conservatively assign 0.1\% as the systematic uncertainty.

\item
{\bf DCS contribution}:
Based on the world-averaged values of the branching fractions, the branching fraction ratios between the known DCS decays and the corresponding CF decays are in the range of (0.2-0.6)\%. Therefore, we take the largest ratio 0.6\% as a conservative estimation of the systematic uncertainty of the DCS effects.

\end{itemize}

The above relative systematic uncertainties are added in quadrature, and a total
of 4.8\%, 6.0\%, 6.6\%, 5.3\% and 6.0\% for the measurements of
${\mathcal B}(D^0\to K^-\pi^+\eta^\prime)$,
${\mathcal B}(D^0\to K^0_S\pi^0\eta^\prime)$,
${\mathcal B}(D^+\to K^0_S\pi^+\eta^\prime)$,
${\mathcal R^0}$, and ${\mathcal R^+}$, respectively, is obtained.

\section{Summary and discussion}

Based on an analysis of an $e^+e^-$ data sample with an integrated luminosity of 2.93 fb$^{-1}$ collected at $\sqrt s= 3.773$~GeV
with the BESIII detector, we measure the branching fractions of hadronic $D$ meson decays to be:
${\mathcal B}(D^0\to K^-\pi^+\eta^\prime)=(6.43 \pm 0.15_{\rm stat.} \pm 0.31_{\rm syst.})\times 10^{-3}$,
${\mathcal B}(D^0\to K^0_S\pi^0\eta^\prime)=(2.52 \pm 0.22_{\rm stat.} \pm 0.15_{\rm syst.})\times 10^{-3}$, and
${\mathcal B}(D^+\to K^0_S\pi^+\eta^\prime)=(1.90 \pm0.17_{\rm stat.} \pm 0.13_{\rm syst.})\times 10^{-3}$.
The measured branching fraction of $D^0\to K^-\pi^+\eta^\prime$ is consistent with the previous result
measured by CLEO~\cite{prd48_4007,pdg2014}, but improved with a factor of 4 in precision.
The branching fractions of $D^0\to K^0_S\pi^0\eta^\prime$ and
$D^+\to K^0_S\pi^+\eta^\prime$ are determined for the first time.

Using the measured branching fractions, we determine the ratios of branching fractions to be
${\mathcal R^0}=0.39\pm0.03_{\rm stat.}\pm0.02_{\rm syst.}$ and
${\mathcal R^+}=0.30\pm0.03_{\rm stat.}\pm0.02_{\rm syst.}$.
${\mathcal R^0}$ agrees well with the value 0.4 predicted by the SIM,
but ${\mathcal R^+}$ significantly deviates from the expected value 0.9.
This deviation may arise from a possible phase difference between two isospin states in the SIM~\cite{lvcd}.
In our analysis, we do not find an obvious $K^{*}$ signal in the $K\pi$ invariant mass distributions, which is
consistent with the predictions of small $D^0\to \bar K^{*0}\eta^\prime$ and $D^+\to K^{*+}\eta^\prime$ contributions~\cite{Ddecay1,Ddecay2,Ddecay3}.

Summing over the branching fractions of $D\to \bar K\pi\eta^\prime$ decays and the other exclusive $D\to\eta^\prime X$ decays in PDG~\cite{pdg2014},
we obtain the sums of the branching fractions of all the exclusive $D^0\to\eta^\prime X$ and $D^+\to\eta^\prime X$ to be
$(3.23\pm0.13)\%$ and $(1.06\pm0.07)\%$, respectively.
They are consistent with the measured inclusive production ${\mathcal B} (D^0\to\eta^\prime X)=(2.48\pm0.27)\%$ and ${\mathcal B} (D^+\to\eta^\prime X)=(1.04\pm0.18)\%$~\cite{prd74_112005} within $2.5\sigma$ and $0.1\sigma$, respectively. This excludes the possibility of additional exclusive $D\to\eta^\prime X$ decay modes with large branching fractions.

\section{Acknowledgements}

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.
The authors are grateful to Fu-Sheng Yu, Jonathan L. Rosner, and Zhizhong Xing for helpful discussions.
This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of
China (NSFC) under Contracts Nos. 11335008, 11425524, 11475123, 11625523, 11635010, 11675200, 11735014, 11775230; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility
Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos.
U1532257, U1532258, U1532101; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS;
INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. Collaborative Research Center
CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03;
Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; The Swedish Research Council; U. S. Department
of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0010504, DE-SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer
Schwerionenforschung GmbH (GSI), Darmstadt.