Flow by Gauss curvature to the orlicz chord Minkowski problem

Abstract

The \(L_p\) chord Minkowski problem based on chord measures and \(L_p\) chord measures introduced firstly by Lutwak et al. (Comm Pure Appl Math 1–54, 2023) is a very important and meaningful geometric measure problem in the \(L_p\) Brunn–Minkowski theory. Xi et al. (Adv Nonlinear Stud 23:20220041, 2023) using variational methods gave a measure solution when \(p > 1\) and \(0<p<1\) in the symmetric case. Recently, Guo et al. (Math Ann 2023. 10.1007/s00208-023-02721-8) also obtained a measure solution for \(0\le p<1\) 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.