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Radial Mean Bodies Are Convex

Dylan Langharst

Abstract

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 unresolved. In this work, we answer this nearly 30-year-old question in the affirmative using Prékopa's theorem. Along the way, we provide a new proof of Keith Ball's theorem on integrals of log-concave functions along rays against the weight $r^{p-1}$ and extend it to $p\in (-1,0)$: if $g$ is an integrable, log-concave function which attains its maximum at the origin, with the origin lying in the interior of its support, 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 finite, positively 1-homogeneous convex function on $\mathbb{R}^n$, i.e. a gauge.

Radial Mean Bodies Are Convex

Abstract

In 1998, R. Gardner and G. Zhang introduced the radial th mean bodies of a convex body for . Furthermore, they established that is convex for , but the convexity of for remained unresolved. In this work, we answer this nearly 30-year-old question in the affirmative using Prékopa's theorem. Along the way, we provide a new proof of Keith Ball's theorem on integrals of log-concave functions along rays against the weight and extend it to : if is an integrable, log-concave function which attains its maximum at the origin, with the origin lying in the interior of its support, then is a finite, positively 1-homogeneous convex function on , i.e. a gauge.
Paper Structure (8 sections, 14 theorems, 106 equations)

This paper contains 8 sections, 14 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Some Properties of Ball Bodies
  3. Proof of Theorem \ref{['t:main']}
  4. Proof of Theorem \ref{['t:ball_expanded']}
  5. Step 1: The Smooth Case.
  6. Step 2: Removing the Regularity Assumptions.
  7. Proof of Theorem \ref{['t:theorem']}
  8. Open Question

Key Result

Proposition 1

Let $f:{\mathbb R}^{n}\times {\mathbb R}^{m}\to {\mathbb R}_+$ be log-concave and integrable. Then, the function on ${\mathbb R}^n$ given by is log-concave.

Theorems & Definitions (24)

  • Proposition 1: Prékopa's theorem, PreL1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Proposition 3: Theorem 5 in Ball88
  • Definition 4
  • Theorem 3
  • Proposition 5
  • Lemma 6
  • proof
  • ...and 14 more