AI 中文总结
本文完成了 Kotrbatý-Mouamine 不等式中等式情形的刻画，证明若凸体满足该等式，则必为单纯形。
AI 中文摘要
Godbersen 长期存在的猜想指出，对于任意凸体 $K \subset \R^n$，其混合体积满足 $V(K[k], -K[n-k])\le \binom{n}{k} \vol(K)$，其中 $k \in \{1, \dots, n - 1\}$，且等号成立当且仅当 $K$ 是 $n$-单纯形。Kotrbatý 和 Mouamine 最近在完全一般性下证明了该不等式，并表明若 $K$ 是达到等号的（全维）多胞形，则它必为单纯形。我们完成了等式情形的刻画：若 $K \subset \R^n$ 是凸体，且对某个 $k \in \{1, \dots, n -1\}$ 有 $V(K[k], -K[n-k]) = \binom{n}{k} \vol(K)$，则 $K$ 是 $n$-单纯形。
英文摘要
Godbersen's long-standing conjecture states that the mixed volumes of any convex body $K \subset \mathbb R^n$ satisfy $V(K[k], -K[n-k])\le \binom{n}{k} \operatorname{vol}(K)$ 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 $K \subset \mathbb R^n$ is a convex body for which $V(K[k], -K[n-k]) = \binom{n}{k} \operatorname{vol}(K)$ for some $k \in \{1, \dots, n -1\}$, then $K$ is an $n$-simplex. By a similar method, we characterize the equality case in the higher-order Godbersen inequality conjectured by Schneider and by Kotrbatý.
Comments14 pages. Version 2: added Theorem 5, and its proof in section 5
