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arXiv · 1904.03255

Rogers-Shepard Type Inequalities for Sections

Michael Roysdon
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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: \[ \mu((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} \mu(K \cap (H+y))? \] We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m) = \binom{n+m}{m}$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\frac{1}{s}\right)$-concave, $0 \leq s < \infty$, and logarithmically concave functions.

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Michael Roysdon. 2019-04-05. Rogers-Shepard Type Inequalities for Sections. https://arxiv.org/abs/1904.03255

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