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Wulff Shapes and a Characterization of Simplices via a Bezout Type Inequality
arXiv preprint arXiv:1801.02675
  • Christos Saroglou, Kent State University
  • Ivan Soprunov, Cleveland State University
  • Artem Zvavitch, Mathematical Sciences Research Institute in Berkeley
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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.

Citation Information
Christos Saroglou, Ivan Soprunov and Artem Zvavitch. "Wulff Shapes and a Characterization of Simplices via a Bezout Type Inequality" arXiv preprint arXiv:1801.02675 (2018) p. 1 - 20
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