Some logarithmic Minkowski inequalities for nonsymmetric convex bodies and related problems

Lewen Ji

doi:10.1186/s13660-020-02314-y

Abstract

In this paper, we show the existence of a solution to an even logarithmic Minkowski problem for p-capacity and prove some analogue inequalities of the logarithmic Minkowski inequality for general nonsymmetric convex bodies involving p-capacity.

Highlights

A convex body in an n-dimensional Euclidean space Rn is a compact convex set that has nonempty interior
The cone-volume measure VK of a convex body K is a Borel measure on the unit sphere Sn–1 defined for a Borel ω ∈ Sn–1 by
The problem asks: What are the necessary and sufficient conditions on a finite Borel measure μ on Sn–1 such that μ is the conevolume measure of a convex body in Rn? In [30], Zhu solved the case of discrete measures whose supports are in general position

Summary

Introduction

A convex body in an n-dimensional Euclidean space Rn is a compact convex set that has nonempty interior. The cone-volume measure VK of a convex body K is a Borel measure on the unit sphere Sn–1 defined for a Borel ω ∈ Sn–1 by In his celebrated paper [17], Jerison solved the Minkowski problem for the capacitary measure, the measure that is the variational functional arising from the electrostatic capacity. The Minkowski problem for p-capacity was posed [9]: Given a finite Borel measure μ on Sn–1, what are the necessary and sufficient conditions on μ so that μ is the pcapacitary measure μp(K, ·) of convex body K in Rn? The measure μ is the L0 p-capacitary measure of an originsymmetric convex body in Rn. we prove the modified logarithmic Minkowski inequality for p-capacity.

Preliminaries

Logarithmic Minkowski type inequality