Wulff shapes and a characterization of simplices via a Bezout type inequality

Abstract

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$V(L_1,\dots ,L_n)V_n(K)\le V(L_1,K[n-1])V(L_2,\dots ,L_n,K).$$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1,\dots ,L_n\subset {\mathbb{R}}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies.

References in corpus (4)

• The Logarithmic Minkowski Problem
• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• Local $L^p$-Brunn-Minkowski inequalities for $p<1$
• Characterization of Simplices via the Bezout Inequality for Mixed volumes

Cited by in corpus (2)

• Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures
• Weighted Minkowski's Existence Theorem and Projection Bodies