The equality case in Godbersen's inequality and its generalizations

Godbersen’s long-standing conjecture states that the mixed volumes of any convex body $K\subset {\mathbb{R}}^n$ satisfy $V(K[k],-K[n-k])\le \left(\genfrac{}{}{0px}{}{n}{k}\right)\mathrm{vol}(K)$ 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 $V(K[k],-K[n-k])=\left(\genfrac{}{}{0px}{}{n}{k}\right)\mathrm{vol}(K)$ 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.