Measuring the convexity of compact sumsets with the Schneider non-convexity index

TL;DR

This paper introduces the Schneider region based on the non-convexity index to analyze sumsets, providing initial characterizations and properties, and compares it to the Lyusternik region for understanding convexity and volume behavior.

Contribution

It defines the Schneider region using the non-convexity index, establishes a lower bound for sumsets in one dimension, and explores its fractional subadditivity and qualitative properties.

Findings

Schneider non-convexity index of sumsets has a best lower bound in 1D.

The Schneider non-convexity index exhibits fractional subadditivity.

The Lyusternik region is not closed for m ≥ 3, revealing new qualitative properties.

Abstract

In recent work, Franck Barthe and Mokshay Madiman introduced the concept of the Lyusternik region, denoted by $\Lambda_{n}(m)$, to better understand volumes of sumsets. They gave a characterization of $\Lambda_{n}(2)$ (the volumes of compact sets in $\mathbb{R}^n$ when at most $m=2$ sets are added together) and proved that Lebesgue measure satisfies a fractional superadditive property. We attempt to imitate the idea of the Lyusternik region by defining a region based on the Schneider non-convexity index function, which was originally defined by Rolf Schneider in 1975. We call this region the Schneider region, denoted by $S_{n}(m)$. In this paper, we will give an initial characterization of the region $S_{1}(2)$ and in doing so, we will prove that the Schneider non-convexity index of a sumset $c(A_1+A_2)$ has a best lower bound in terms of $c(A_1)$ and $c(A_2)$. We will pose some open questions about extending this lower bound to higher dimensions and large sums. We will also show that, analogous to Lebesgue measure, the Schneider non-convexity index has a fractional subadditive property. Regarding the Lyusternik region, we will show that when the number of sets being added is $m\geq3$, that the region $\Lambda_{n}(m)$ is not closed, proving a new qualitative property for the region.