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arXiv preprint arXiv:1801.02675


Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes V(L1,...,Ln)Vn(K) ≤ V(L1,K[n-1])V(L2,..., 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 L1, ..., Ln ⊂ Rn, 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.


The third author is supported in part by U.S. National Science Foundation Grant DMS-1600753, this material is based upon work supported by the US National Science Foundation under Grant DMS- 1440140 while the third author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.



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