Rogers-Shepard Type Inequalities for Sections

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...

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Bibliographic Details
Published in:arXiv.org 2019-09
Main Author: Roysdon, Michael
Format: Article
Language:English
Subjects:
Inequalities
Inequality
Online Access:Get full text
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Summary: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.
ISSN:2331-8422