Mathematics

CLC number: O186.5

The L_p,s-Gaussian-Minkowski Problem on Even Measures

偶测度的L_p,s-高斯-Minkowski问题

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received: 1 March 2025

Abstract

In this paper, we introduce the concept of the $L_{p,s}$-Gaussian surface area measure of a convex body in $n$-dimensional Euclidean space ${\mathbb{R}}^n$ and formulate the corresponding $L_{p,s}$-Gaussian-Minkowski problem: Given a finite Borel measure $\mu$ on $S^{n-\mathrm{1}}$, what are the necessary and sufficient conditions for the existence of a convex body whose $L_{p,s}$-Gaussian surface area measure equals measure $\mu$? Furthermore, we present a solution to the $L_{p,s}$-Gaussian-Minkowski problem for the case of even measures.

摘要

本文引入了$n$维欧氏空间${\mathbb{R}}^n$中凸体的$L_{p,s}$-高斯表面积测度的概念， 并提出了相应的$L_{p,s}$-高斯-Minkowski问题， 即当一个$S^{n-\mathrm{1}}$上的有限Borel测度$\mu$满足什么充要条件时保证存在一个凸体， 使得该凸体的$L_{p,s}$-高斯表面积测度等于测度$\mu$。 此外， 我们给出了关于偶测度的$L_{p,s}$-高斯-Minkowski问题的一个解。