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

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 $m\ge 3$, that the region ${\Lambda }_n(m)$ is not closed, proving a new qualitative property for the region.