Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

TL;DR

This paper proves new geometric inequalities in dual Brunn-Minkowski theory, including a conjectured Brunn-Minkowski inequality for dual quermassintegrals and a reverse isoperimetric inequality highlighting the cube's extremal property.

Contribution

It establishes the first proof of Lutwak's conjectured Brunn-Minkowski inequality for dual quermassintegrals and introduces a reverse inequality identifying the cube as extremal.

Findings

Proof of Lutwak's conjectured inequality

Cube maximizes dual quermassintegrals in John’s position

Extension of Ball's volume ratio inequality

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.