Equality conditions for the fractional superadditive volume inequalities

While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in ${\mathbb{R}}^n$. In doing this they proved a fractional generalization of the Brunn-Minkowski-Lyusternik (BML) inequality in dimension $n=1$. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition $(\mathcal{G},\beta )$ and nonempty sets $A_1,\dots ,A_m\subseteq \mathbb{R}$, equality holds iff for each $S\in \mathcal{G}$, the set ${\sum }_{i\in S}{A}_i$ is an interval. In the case of dimension $n\ge 2$ we will show that equality can hold if and only if the set ${\sum }_{i=1}^m{A}_i$ has measure $0$.