On a normed version of a Rogers-Shephard type problem

Lángi, Zsolt (2016) On a normed version of a Rogers-Shephard type problem. ISRAEL JOURNAL OF MATHEMATICS. pp. 1-15. ISSN 0021-2172

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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 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.

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