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

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.