Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem

Abstract

This paper deals with processes of combining convex bodies in Euclidean n-space which are, in a sense, dual to the process of Minkowski addition and some of its generalizations.All the convex bodies considered will have a common interior point Q. Variables x and y denote vectors drawn from Q; we shall speak of their terminal points as the points x and y. Unit vectors will be denoted by u; ||x|| signifies the length of x. Convex bodies will be symbolized by K with distinguishing marks. ∂K means the boundary of K. λK will mean the image of K under a homothetic transformation in the ratio λ : 1. The centre of the homothety will always be Q.

The Brunn-Minkowski inequality

“…It was found by Firey [57] for convex bodies and p ≤ −1. The general inequality forms part of Lutwak's highly successful dual Brunn-Minkowski theory, in which the intersections of star bodies with subspaces replace the projections of convex bodies onto subspaces in the classical theory; see, for example, [66].…”

Mean cross-section measures of harmonic means of convex bodies

“…Our second tool, which we shall refer to as the projection lemma, was established in [2]. Let K* denote the projection of K onto a fixed, m-dimensional, linear subspace E m through Q for 1 ^ m < n. We have Since E m contains Q and the polar reciprocation is with respect to sphere E centred at Q, in forming K* the order of operations is immaterial.…”

Inequalities on asymmetric Lp-harmonic radial bodies

“…The L p -harmonic radial combination of convex bodies was first investigated by Firey (see [3,4]) and was extended to star bodies by Lutwak (see [15]), which played an important role in L p Brunn-Minkowski theory. For the further researches of L p -harmonic radial combination, also see ( [1,21,30]).…”