Gr\"unbaum's inequality for sections

Abstract

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.