On the equality conditions of the Brunn-Minkowski theorem

Abstract:

This article describes a new proof of the equality condition for the Brunn-Minkowski inequality. The Brunn-Minkowski Theorem asserts that, for compact convex sets $K,L \subseteq \mathbb {R}^n$, the $n$-th root of the Euclidean volume $V_n$ is concave with respect to Minkowski combinations; that is, for $\lambda \in [0,1]$, \[ V_{n}((1-\lambda )K + \lambda L)^{1/n} \geq (1-\lambda ) V_{n}(K)^{1/n} + \lambda V_{n}(L)^{1/n}.\] The equality condition asserts that if $K$ and $L$ both have positive volume, then equality holds for some $\lambda \in (0,1)$ if and only if $K$ and $L$ are homothetic.