The Characteristic Properties of the Minimal Lp-Mean Width

Tongyi Ma

doi:10.1155/2017/2943073

Abstract

Giannopoulos proved that a smooth convex body K has minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn-1, is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal Lp-mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal Lp-mean width of convex bodies and prove the existence and uniqueness of the minimal Lp-mean width in its SL(n) images. In addition, we establish a characterization of the minimal Lp-mean width, conclude the average Mp(K) with a variation of the minimal Lp-mean width position, and give the condition for the minimum position of Mp(K).

Highlights

Let Ln(Rn) denote the space of linear operators from Rn to Rn and SL(n) = {T ∈ Ln(Rn) : det(T) = 1}
Recall that the width of K in the direction of u ∈ Sn−1 is defined by w(K, u) = hK(u) + hK(−u), where hK(x) is the support function of convex K
In [1], the authors show that a smooth enough convex body is in minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn−1, is isotropic

Summary

Introduction

In [1], the authors show that a smooth enough convex body (that is, hK is twice continuously differentiable) is in minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn−1, is isotropic. Let K be a convex body in Rn and p ≥ 1; the Lp-width of K in the direction of u ∈ Sn−1 is defined by wp(K, u) = hKp (u) + hKp (−u). The following isotropic characterization of the minimal Lp-mean width position was proved in [2]. Please see the section for above interrelated notations, definitions, and their background materials

Preliminaries

The Continuity of the Lp-Mean Width

The Minimal Lp-Mean Width

The Characterization of the Minimal Lp-Mean Width