Volumes of subset Minkowski sums and the Lyusternik region

TL;DR

This paper investigates the Lyusternik region, the set of possible volumes of Minkowski subset sums, establishing a fractional Brunn-Minkowski-Lyusternik inequality in one dimension and a variant in higher dimensions.

Contribution

It introduces the Lyusternik region and proves a fractional inequality related to Minkowski sums, extending previous conjectures and results.

Findings

Fractional Brunn-Minkowski-Lyusternik inequality holds in dimension 1

A variant of the inequality holds in higher dimensions

Initial steps towards describing 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.