While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in R. 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 (G,β) and nonempty sets A1,⋯,A⊆R, equality holds iff for each S∈G, the set ∑A is an interval. In the case of dimension n≥2 we will show that equality can hold if and only if the set ∑i=1A has measure 0. © The Author(s) 2024.