The Generalized Gaussian Minkowski Problem

2024/02/22 by Jiaqian Liu, Liu, Jiaqian, Shengyu Tang +1
Computer Science · Mathematics · #35J96 #52A38 #52A40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geochemistry and Geologic Mapping #Morphological variations and asymmetry #Probability (math.PR) #Statistical and numerical algorithms

paper · pdf · doi:10.48550/arxiv.2402.14237

openalex publication_date 2024/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article delves into the L_p Minkowski problem within the framework of generalized Gaussian probability space. This type of probability space was initially introduced in information theory through the seminal works of Lutwak, Yang, and Zhang [49,50], as well as by Lutwak, Lv, Yang, and Zhang [45]. The primary focus of this article lies in examining the existence of this problem. While the variational method is employed to explore the necessary and sufficient conditions for the existence of the normalized Minkowski problem when p ∈ ℝ ∖ \0\, our main emphasis is on the existence of the generalized Gaussian Minkowski problem without the normalization requirement, particularly in the smooth category for p ≥ 1.

Related

• The Minkowski problem in the Gaussian probability space
• The generalized Gaussian log-Minkowski problem
• The L_p-Gaussian Minkowski problem
• Flow by Gauss curvature to the L_p-Gaussian Minkowski problem
• Existence of Solutions to L_p-Gaussian Minkowski problem