Dual Quermassintegral Differences for Intersection Body

Lingzhi Zhao,Jun Yuan

doi:10.4236/apm.2014.410061

Abstract

In this paper, we introduce the concept of dual quermassintegral differences. Further, we give the dual Brunn-Minkowski inequality and dual Minkowski inequality for dual quermassintegral differences for mixed intersection bodies.

Highlights

The projection body was introduced in 1934 by Minkowski [1]
The intersection operator and the class of intersection bodies were introduced in 1988 by Lutwak [2], who found a close connection between those bodies and famous Busemann-Petty problem
In [2], Lutwak presented the mysterious duality between projection and intersection bodies

Summary

Introduction

The projection body was introduced in 1934 by Minkowski [1]. The research on the projection body has attracted much attention. In [2], Lutwak presented the mysterious duality between projection and intersection bodies. For convex bodies K and L , let ΠK and Π1 ( K, L) denote the projection body of K and mixed projection body of K and L , respectively. In [7], Lutwak established the following Brunn-Minkowski inequality for projection body and Minkowski inequality for mixed projection body: Theorem A. For star bodies K and L , let IK and I1 ( K, L) denote the intersection body of K and mixed intersection body of K and L , respectively. In [9], Zhao et al established the following dual Brunn-Minkowski inequality for intersection body and dual Minkowski inequality for mixed intersection body: Theorem E. Please see the section for related definitions and notations

Definitions and Notations

Inequalities for Dual Quermassintegral Differences