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

In recent work, Franck Barthe and Mokshay Madiman introduced the concept of the Lyusternik region, denoted by Λₙ(m), to better understand volumes of sumsets. They gave a characterization of Λₙ(2) (the volumes of compact sets in Rⁿ 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 Sₙ(m). In this paper, we will give an initial characterization of the region S₁(2) and in doing so, we will prove that the Schneider non-convexity index of a sumset c(A₁+A₂) has a best lower bound in terms of c(A₁) and c(A₂). 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≥3, that the region Λₙ(m) is not closed, proving a new qualitative property for the region.