Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gr\"unbaum's inequality for sections   We show \begin{align*} \frac{ \int_{E \cap \theta^+} f(x) dx }{ \int_E f(x)
dx }
  \geq \left(\frac{k \gamma+1}{(n+1) \gamma+1}\right)^{\frac{k
\gamma+1}{\gamma}} \end{align*} for all $k$-dimensional subspaces
$E\subset\mathbb{R}^n$, $\theta\in E\cap S^{n-1}$, and all $\gamma$-concave
functions $f:\mathbb{R}^n\rightarrow [0,\infty)$ with $\gamma >0$, $0<
\int_{\mathbb{R}^n} f(x)\, dx <\infty$, and $\int_{\mathbb{R}^n} x f(x)\, dx$
at the origin $o\in\mathbb{R}^n$. Here, $\theta^+ := \lbrace x\, : \, \langle
x,\theta\rangle \geq 0 \rbrace$. As a consequence of this result, we get the
following generalization of Gr\"unbaum's inequality: \begin{align*} \frac{
\mbox{vol}_k(K\cap E\cap\theta^+) }{ \mbox{vol}_k(K\cap E) } \geq \left(
\frac{k}{n+1} \right)^k \end{align*} for all convex bodies
$K\subset\mathbb{R}^n$ with centroid at the origin, $k$-dimensional subspaces
$E\subset\mathbb{R}^n$, and $\theta\in E\cap S^{n-1}$. The lower bounds in both
of our inequalities are the best possible, and we discuss the equality
conditions.

14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg