On a normed version of a Rogers-Shephard type problem

TL;DR

This paper introduces a normed variant of the Rogers-Shephard problem, determining extremal volume values of translation bodies in the plane for various volume types and exploring higher dimensions and symmetric bodies.

Contribution

It extends the classical Rogers-Shephard problem to a normed setting and provides explicit solutions in the planar case for multiple volume measures.

Findings

Explicit extremal volume values for planar convex bodies

Extension of the problem to higher dimensions

Results for centrally symmetric convex bodies

Abstract

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of $n$-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities, with one exception, for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.