Reverse Alexandrov--Fenchel inequalities for zonoids

Abstract

The Alexandrov--Fenchel inequality bounds from below the square of the mixed volume $V(K_1,K_2,K_3,\ldots,K_n)$ of convex bodies $K_1,\ldots,K_n$ in $\mathbb{R}^n$ by the product of the mixed volumes $V(K_1,K_1,K_3,\ldots,K_n)$ and $V(K_2,K_2,K_3,\ldots,K_n)$. As a consequence, for integers $\alpha_1,\ldots,\alpha_m\in\mathbb{N}$ with $\alpha_1+\cdots+\alpha_m=n$ the product $V_n(K_1)^{\frac{\alpha_1}{n}}\cdots V_n(K_m)^{\frac{\alpha_m}{n}} $ of suitable powers of the volumes $V_n(K_i)$ of the convex bodies $K_i$, $i=1,\ldots,m$, is a lower bound for the mixed volume $V(K_1[\alpha_1],\ldots,K_m[\alpha_m])$, where $\alpha_i$ is the multiplicity with which $K_i$ appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume $V(K_1[\alpha_1],\ldots,K_m[\alpha_m])$ in terms of the product of the intrinsic volumes $V_{\alpha_i}(K_i)$, for $i=1,\ldots,m$. The case where $m=2$, $\alpha_1=1$, $\alpha_2=n-1$ has recently been settled by the present authors (2020). The case where $m=3$, $\alpha_1=\alpha_2=1$, $\alpha_3=n-2$ has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that $K_2$ is a zonoid and $K_3$ is the Euclidean unit ball. The case where $\alpha_2=\cdots=\alpha_m=1$, $K_1$ is the unit ball and $K_2,\ldots,K_m$ are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov--Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.