Convexity of Radial Mean Bodies via an extension of Ball's Bodies

TL;DR

This paper extends a classical theorem on log-concave functions to the regime p in (-1,0), proving convexity of radial pth mean bodies for all p > -1, and provides new proofs and insights.

Contribution

It extends Ball's theorem to negative p, proving convexity of radial pth mean bodies for all p > -1, resolving a long-standing open problem.

Findings

Extended Ball's theorem to p in (-1,0)

Proved convexity of radial pth mean bodies for all p > -1

Provided a new proof of the original regime p > 0

Abstract

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable log-concave function which attains its maximum at the origin, then \[ x\mapsto \left(\frac{p}{g(o)}\int_{0}^{\infty}r^{p-1}(g(rx)-g(o))\mathrm{d}\,r\right)^{-\frac{1}{p}} \] is a positively 1-homogeneous convex function on $\mathbb{R}^n$. Our approach also provides a new proof of the original regime $p> 0$. The argument is based on a reduction to a two-dimensional inequality derived from Pr\'ekopa's theorem, which may be of independent interest. As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative. In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K\subset \mathbb{R}^n$ for $p>-1$. Furthermore, they established that $R_p K$ is convex for $p\geq 0$, but the convexity of $R_p K$ for $p\in (-1,0)$ remained open. We prove that $R_p K$ is convex for all $p>-1$.