Rogers-Shepard Type Inequalities for Sections

Abstract

In this paper we address the following question: given a measure $\mu$ on ${\mathbb{R}}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H\subset {\mathbb{R}}^n$ and any convex body $K\subset {\mathbb{R}}^n$, the following sectional Rogers-Shephard type inequality holds: \[ μ((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} μ(K \cap (H+y))? \] We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m)=\left(\genfrac{}{}{0ex}{}{n+m}{m}\right)$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\frac{1}{s}\right)$-concave, $0\le s<\infty$, and logarithmically concave functions.

28 pages