Centrally symmetric convex bodies and sections having maximal quermassintegrals

TL;DR

This paper investigates conditions under which convex bodies have sections with maximal quermassintegrals, extending previous symmetry results to surface area and lower-dimensional measures for small perturbations of the Euclidean ball.

Contribution

It extends symmetry characterizations to surface area and quermassintegrals for convex bodies near the Euclidean ball, using local perturbation analysis.

Findings

Maximal surface area sections characterize symmetric bodies.

Local perturbations of the Euclidean ball preserve maximal section properties.

Results apply to small $C^2$ and $C^3$ perturbations.

Abstract

Let $d \ge 2$, and let $K \subset {\Bbb{R}}^d$ be a convex body containing the origin $0$ in its interior. In a previous paper we have proved the following. The body $K$ is $0$-symmetric if and only if the following holds. For each $\omega \in S^{d-1}$, we have that the $(d-1)$-volume of the intersection of $K$ and an arbitrary hyperplane, with normal $\omega$, attains its maximum if the hyperplane contains $0$. An analogous theorem, for $1$-dimensional sections and $1$-volumes, has been proved long ago by Hammer (\cite{H}). In this paper we deal with the ($(d-2)$-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small $C^2$-perturbations, or $C^3$-perturbations of the Euclidean unit ball, respectively.