Weighted Berwald's Inequality

Abstract

The inequality of Berwald is a reverse-Hölder 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.

Discussed weighted spectral mean bodies, extended the results to p in (-1,0) (v3); streamlined proof of main theorem (v4); Added E. Putterman as co-author, obtained equality conditions for radial mean body set-inclusions for s-concave measures (v5); changed name from "Generalizations of Berwald's Inequality to Measures" (v6); To appear in Indiana University Mathematics Journal (v7)

Cited by in corpus (1)

• On the volume of sums of anti-blocking bodies