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arXiv · 2610.04527

Equality case in the Kotrbatý-Mouamine inequality

Omer Friedland·Eli Putterman·Maud Szusterman
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Abstract

Godbersen's long-standing conjecture states that the mixed volumes of any convex body $K \subset \R^n$ satisfy $V(K[k], -K[n-k])\le \binom{n}{k} \vol(K)$ for any $k \in \{1, \dots, n - 1\}$, with equality if and only if $K$ is an $n$-simplex. Kotrbatý and Mouamine recently proved the inequality in full generality, and showed that if $K$ is a (full-dimensional) polytope attaining equality, it must be a simplex. We complete the characterization of the equality case : if $K \subset \R^n$ is a convex body for which $V(K[k], -K[n-k]) = \binom{n}{k} \vol(K)$ for some $k \in \{1, \dots, n -1\}$, then $K$ is an $n$-simplex.

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Omer Friedland, Eli Putterman, Maud Szusterman. 2026-10-03. Equality case in the Kotrbatý-Mouamine inequality. https://arxiv.org/abs/2610.04527

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