On the Lp Gaussian Minkowski problem

TL;DR

This paper investigates the existence and uniqueness of solutions to the Lp Gaussian Minkowski problem across different parameter ranges, expanding understanding of convex geometric analysis with Gaussian measures.

Contribution

It establishes existence results for symmetric and asymmetric solutions for various p-values and proves uniqueness of smooth solutions for p>n without volume restrictions.

Findings

Existence of solutions for p ≤ 0 and p ≥ 1.

Uniqueness of smooth solutions for p > n.

Solutions exist without volume restrictions for p > n.

Abstract

Existence of symmetric (resp. asymmetric) solutions to the $L_p$ Gaussian Minkowski problem for $p\leq 0$ (resp. $p\geq 1$) will be provided. Moreover, existence and uniqueness of smooth solutions to the problem for $p>n$ will also be proved without the restriction that the Gaussian volumes of convex bodies are not less than one-second.