  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 $n\geq2$ we will show that equality can hold if and only if the set
$\sum_{i=1}^{m}A_i$ has measure $0$.
