The role of the Rogers-Shephard inequality in the characterization of the difference body

TL;DR

This paper explores how the Rogers-Shephard inequality characterizes the difference body operator among convex bodies, highlighting its unique role in geometric inequalities and operator properties.

Contribution

It demonstrates that the difference body operator uniquely satisfies a Rogers-Shephard type inequality among continuous, GL(n)-covariant operators.

Findings

The difference body operator is the only one satisfying a Rogers-Shephard inequality under certain conditions.

All continuous, GL(n)-covariant operators satisfy a Brunn-Minkowski type inequality.

The study clarifies the interplay between inequalities and operator properties in convex geometry.

Abstract

The difference body operator enjoys different characterization results relying on its basic properties such as continuity, SL(n)-covariance, Minkowski valuation or symmetric image. The Rogers-Shephard and the Brunn-Minkowski inequalities provide upper and lower bounds for the volume of the difference body in terms of the volume of the body itself. In this paper we aim to understand the role of the Rogers-Shephard inequality in characterization results of the difference body and, at the same time, to study the interplay among the different properties. Among others, we prove that the difference body operator is the only continuous and GL(n)-covariant operator from the space of convex bodies to the origin-symmetric ones which satisfies a Rogers-Shephard type inequality while every continuous and GL(n)-covariant operator satisfies a Brunn-Minkowski type inequality.