Weighted Berwald's Inequality

TL;DR

This paper extends Berwald's inequality to measures with concavity properties, providing new equality conditions and applications to convex geometry, including bounds on intersections and concepts like radial means bodies.

Contribution

It generalizes Berwald's inequality to s-concave measures and establishes new equality conditions, enhancing understanding of convex geometric inequalities.

Findings

Proved Berwald's inequality for s-concave measures.

Derived new equality conditions for classical Berwald's inequality.

Extended bounds for convex body intersections and related geometric concepts.

Abstract

The inequality of Berwald is a reverse-H\"older like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions with respect to measures that have some concavity conditions, e.g. $s$-concave measures, $s\in \mathbb{R}.$ We also obtain equality conditions; in particular, this provides a new proof for the equality conditions of the classical inequality of Berwald. As applications, we generalize a number of classical bounds for the measure of the intersection of a convex body with a half-space and also the concept of radial means bodies and the projection body of a convex body.