Volumes of subset Minkowski sums and the Lyusternik region

Abstract

We begin a systematic study of the region of possible values of the volumes of Minkowski subset sums of a collection of $M$ compact sets in ${\mathbb{R}}^d$, which we call the Lyusternik region, and make some first steps towards describing it. Our main result is that a fractional generalization of the Brunn-Minkowski-Lyusternik inequality conjectured by Bobkov et al. (2011) holds in dimension 1. Even though Fradelizi et al. (2016) showed that it fails in general dimension, we show that a variant does hold in any dimension.

References in corpus (3)

• Dimensional behaviour of entropy and information
• Cores of Cooperative Games in Information Theory
• Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

Cited by in corpus (3)

• On the volume of the Minkowski sum of zonoids
• Bernoulli sums and Rényi entropy inequalities
• Sumset estimates in convex geometry