Uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$

TL;DR

This paper proves the uniqueness of solutions to the logarithmic Minkowski problem in three-dimensional space without symmetry assumptions, under conditions where the measure's density is close to 1 in a specific norm, also impacting related curvature flow solutions.

Contribution

It establishes the first uniqueness result for the 3D logarithmic Minkowski problem without symmetry, under near-constant density conditions.

Findings

Uniqueness of solutions in $\, ext{R}^3$ for the logarithmic Minkowski problem.

Implication of uniqueness for self-similar solutions to anisotropic Gauss curvature flow.

Results hold when the measure's density is close to 1 in $C^{eta}$ norm.

Abstract

In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$ without symmetry condition, provided the density of the measure is close to $1$ in $C^{\alpha}$ norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in $\mathbb{R}^3$ when the speed function is $C^{\alpha}$ close to a positive constant.