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While studying set function properties of Lebesgue measure, Franck Barthe andMokshay Madiman proved that Lebesgue measure is fractionally superadditive oncompact sets in $\mathbb{R}^n$. In doing this they proved a fractionalgeneralization of the Brunn-Minkowski-Lyusternik (BML) inequality in dimension$n=1$. In this article a complete characterization of the equality conditionsfor the fractional BML inequality in one-dimension will be given. It will beshown that aside from some trivial cases, that for a fractional partition$(\mathcal{G},\beta)$ and non-empty compact sets$A_1,\dots,A_m\subset\mathbb{R}$ equality holds if and only if for each$S\in\mathcal{G}$ the set $\sum_{i\in S}A_i$ is either an interval withpositive measure or consists of exactly one point.

Equality conditions for the fractional Brunn-Minkowski-Lyusternik inequality in one-dimension