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

Abstract

While studying set function properties of Lebesgue measure, Franck Barthe and Mokshay 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 article a complete characterization of the equality conditions for the fractional BML inequality in one-dimension will be given. It will be shown 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 with positive measure or consists of exactly one point