Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem

TL;DR

This paper proves the uniqueness of solutions to a specific geometric problem involving Gaussian measures and Minkowski problems in higher dimensions, for a range of parameters, without assuming bodies are centered at the origin.

Contribution

It establishes the uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem for certain $p$ values without the origin-centred assumption.

Findings

Uniqueness holds for $-(n+1)<p<-1$ in $ +1$ dimensions.

No need for origin-centred convex bodies in the proof.

Extends understanding of Gaussian Minkowski problems.

Abstract

The uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem in $\mathbb{R}^{n+1}$ is established when $-(n+1)<p<-1$ with $n\geq 1$, without requiring the origin-centred assumption on convex bodies.