Document Type

Article

Publication Date

12-1-2016

Publication Title

International Mathematics Research Notices

Abstract

In this paper we consider the following analog of Bezout inequality for mixed volumes: V(P1,…,Pr,Δn−r)Vn(Δ)r−1≤∏i=1rV(Pi,Δn−1) for 2≤r≤n. We show that the above inequality is true when Δ is an n-dimensional simplex and P1,…,Pr are convex bodies in Rn. We conjecture that if the above inequality is true for all convex bodies P1,…,Pr, then Δ must be an n-dimensional simplex. We prove that if the above inequality is true for all convex bodies P1,…,Pr, then Δ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to Δ), which confirms the conjecture when Δ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.

Comments

The first author is supported in part by NSA Grant H98230-13-1-0279. The second author is supported in part by U.S. National Science Foundation Grant DMS-1101636 and by the Simons Foundation.

DOI

10.1093/imrn/rnv390

Version

Postprint

Issue

23

Available for download on Friday, November 16, 2018

Included in

Mathematics Commons

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