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The Logarithmic Minkowski Problem

Károly J. Böröczky, Erwin Lutwak, Deane Yang, Gaoyong Zhang

TL;DR

The paper solves the even logarithmic Minkowski problem by identifying the subspace concentration condition as the sharp existence criterion: a nonzero finite even measure on the unit sphere is the cone-volume measure of an origin-symmetric convex body if and only if the measure satisfies the subspace concentration condition. The authors develop a variational framework on Aleksandrov/Wulff bodies, use a logarithmic functional and a Monge-Ampère type relation, and prove existence via a direct variational argument plus compactness (Blaschke) and dimension-reduction arguments. This work extends the theory around cone-volume measures and connects to the $L_p$-Minkowski problem, providing a rigorous route to measure-data solvability that does not follow from functional-data approximations. The results have implications for the geometry of finite-dimensional Banach spaces and affine isoperimetric inequalities, strengthening the link between convex geometry and functional-analytic structures.

Abstract

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

The Logarithmic Minkowski Problem

TL;DR

The paper solves the even logarithmic Minkowski problem by identifying the subspace concentration condition as the sharp existence criterion: a nonzero finite even measure on the unit sphere is the cone-volume measure of an origin-symmetric convex body if and only if the measure satisfies the subspace concentration condition. The authors develop a variational framework on Aleksandrov/Wulff bodies, use a logarithmic functional and a Monge-Ampère type relation, and prove existence via a direct variational argument plus compactness (Blaschke) and dimension-reduction arguments. This work extends the theory around cone-volume measures and connects to the -Minkowski problem, providing a rigorous route to measure-data solvability that does not follow from functional-data approximations. The results have implications for the geometry of finite-dimensional Banach spaces and affine isoperimetric inequalities, strengthening the link between convex geometry and functional-analytic structures.

Abstract

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

Paper Structure

This paper contains 7 sections, 11 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The variational method
  4. A minimization problem
  5. Necessary conditions for existence
  6. Minimizing the logarithmic functional
  7. Existence for the even logarithmic Minkowski problem

Key Result

Theorem 1.1

A non-zero finite even Borel measure on the unit sphere $S^{n-1}$ is the cone-volume measure of an origin-symmetric convex body in $\mathbb R^n$ if and only if it satisfies the subspace concentration condition.

Theorems & Definitions (21)

  • Definition
  • Theorem 1.1
  • Lemma 3.1
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • proof
  • Lemma 6.1
  • ...and 11 more