Equivalence of Some Affine Isoperimetric Inequalities

Wuyang Yu

doi:10.1155/2009/981258

Abstract

We establish the equivalence of some affine isoperimetric inequalities which include the -Petty projection inequality, the -Busemann-Petty centroid inequality, the dual -Petty projection inequality, and the dual -Busemann-Petty inequality. We also establish the equivalence of an affine isoperimetric inequality and its inclusion version for -John ellipsoids.

Highlights

In the recent years, the Lp-analogs of the projection bodies and centroid bodies have received considerable attentions 1–7
Corresponding to Lutwak et al.’s work, Yu et al 9 proved that there is a family of dual Lp-John ellipsoids, EpK, which can be associated with a fixed convex body K: if K contains the origin in its interior and p > 0, among all origin-centered ellipsoids E, the unique ellipsoid EpK solves the constrained maximization problem: V EpK
If K is a convex body in Rn that contains the origin in its interior, and 1 ≤ p, ωn 2n

Summary

Introduction

The Lp-analogs of the projection bodies and centroid bodies have received considerable attentions 1–7. Lutwak et al established the Lp-analog of the Petty projection inequality 4 It states that if K is a convex body in Rn, for 1 ≤ p < ∞,. Lutwak et al 8 proved that there is a family of Lp-John ellipsoids, EpK, which can be associated with a fixed convex body K: if K contains the origin in its interior and p > 0, among all origin-centered ellipsoids E, the unique ellipsoid EpK solves the constrained maximization problem:. Corresponding to Lutwak et al.’s work, Yu et al 9 proved that there is a family of dual Lp-John ellipsoids, EpK, which can be associated with a fixed convex body K: if K contains the origin in its interior and p > 0, among all origin-centered ellipsoids E, the unique ellipsoid EpK solves the constrained maximization problem:

EpK max

Notations and Background Materials

Proof of the Results

K ωn 2nV EpK