  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.
