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A note on Bézout type inequalities for mixed volumes and Minkowski sums

Cheikh Saliou Ndiaye

Abstract

In this note, we study B{é}zout type inequalities for mixed volume and Minkowski sum of convex bodies in R n. We first give a new proof and we extend inequalities of Jian Xiao on mixed discriminants. Then, we use mass transport method to deduce some B{é}zout type inequalities for mixed volumes. Finally, we apply these inequalities to obtain B{é}zout type inequalities for Minkowski sums.

A note on Bézout type inequalities for mixed volumes and Minkowski sums

Abstract

In this note, we study B{é}zout type inequalities for mixed volume and Minkowski sum of convex bodies in R n. We first give a new proof and we extend inequalities of Jian Xiao on mixed discriminants. Then, we use mass transport method to deduce some B{é}zout type inequalities for mixed volumes. Finally, we apply these inequalities to obtain B{é}zout type inequalities for Minkowski sums.
Paper Structure (4 sections, 8 theorems, 81 equations)

This paper contains 4 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Xiao type inequalities
  3. Bézout inequality for mixed volumes.
  4. Bézout inequality for Minkowski sums.

Key Result

Theorem 1

Let $m\geq 1, n\geq 2$ and $i_1,\dots,i_m\geq 0$ be integers such that $|i|:=i_1+\dots+i_m\leq n$. Then, for any positive semi-definite symmetric matrices $A,B_1,\dots,B_m,$$M_1,\dots,M_{n-|i|}$,

Theorems & Definitions (11)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Corollary 4
  • Remark 5
  • Theorem 6
  • Lemma 7
  • Theorem 8
  • Remark 9
  • Theorem 10
  • ...and 1 more