Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

TL;DR

This paper extends the $L_p$ chord Minkowski problem to the Orlicz setting, proving the existence of smooth solutions using Gauss curvature flows, thus broadening the scope of geometric measure problems.

Contribution

It introduces the Orlicz chord Minkowski problem, generalizing the $L_p$ case, and establishes existence results for smooth solutions via curvature flow methods.

Findings

Established existence of smooth solutions for the Orlicz chord Minkowski problem.

Generalized the $L_p$ chord Minkowski problem to the Orlicz setting.

Applied Gauss curvature flow techniques to solve the problem.

Abstract

The $L_p$ chord Minkowski problem based on Chord measures and $L_p$ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the $L_p$ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when $p > 1$ and $0<p<1$ in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for $0\leq p<1$ by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the $L_p$ chord Minkowski problem by replacing $p$ with a fixed continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.