Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Volumes of subset Minkowski sums and the Lyusternik region

Franck Barthe, Mokshay Madiman

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.

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 compact sets in , 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.
Paper Structure (12 sections, 9 theorems, 65 equations, 2 figures)

This paper contains 12 sections, 9 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem \ref{['thm:FSA']}
  4. A reduction to regular fractional partitions
  5. Starting with examples
  6. Proof for the real line
  7. Extension to higher dimensions
  8. Other proofs
  9. Cartesian products
  10. Proof for the integers
  11. Concluding remarks and open questions
  12. The relevance of fractional superadditivity

Key Result

Theorem 1

For any $d\in{\mathbb N}^*$ and $M\geq 3$,

Figures (2)

  • Figure 1: Leave-one-out partition on $\{1,2,3\}$
  • Figure 2: A 3-regular partition of $\{1,2,3,4,5\}$

Theorems & Definitions (19)

  • Conjecture 1
  • Definition
  • proof
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 9 more