Grünbaum's inequality for sections

Abstract

We show \begin{align*} \frac{ \int_{E \cap θ^+} f(x) dx }{ \int_E f(x) dx } \geq \left(\frac{k γ+1}{(n+1) γ+1}\right)^{\frac{k γ+1}γ} \end{align*} for all $k$-dimensional subspaces $E\subset {\mathbb{R}}^n$, $\theta \in E\cap S^{n-1}$, and all $\gamma$-concave functions $f:{\mathbb{R}}^n\to [0,\infty )$ with $\gamma >0$, $0<{\int }_{{\mathbb{R}}^n}f(x)dx<\infty$, and ${\int }_{{\mathbb{R}}^n}xf(x)dx$ at the origin $o\in {\mathbb{R}}^n$. Here, ${\theta }^{+}:=\{x:⟨x,\theta ⟩\ge 0\}$. As a consequence of this result, we get the following generalization of Grünbaum's inequality: \begin{align*} \frac{ \mbox{vol}_k(K\cap E\capθ^+) }{ \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.

17 pages, 3 figures