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Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

Shay Sadovsky, Gaoyong Zhang

Abstract

This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The second, generalizing Ball's volume ratio inequality, is a reverse isoperimetric inequality: among all origin-symmetric convex bodies in John's position, the cube maximizes the dual quermassintegrals.

Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

Abstract

This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The second, generalizing Ball's volume ratio inequality, is a reverse isoperimetric inequality: among all origin-symmetric convex bodies in John's position, the cube maximizes the dual quermassintegrals.

Paper Structure

This paper contains 4 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Brunn-Minkowski inequality for dual quermassintegrals
  3. Reverse isoperimetric inequality for dual quermassintegrals
  4. Problems

Key Result

Theorem 2

Let $K,L$ be origin-symmetric convex bodies in $\mathbb R^n$. Then, for $0< q \le n$,

Theorems & Definitions (13)

  • Conjecture 1: Lutwak
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof : Proof of Theorem \ref{['thm:BM']}
  • Theorem 7: schechtman1995concentration
  • ...and 3 more