Flow by Gauss curvature to the orlicz chord Minkowski problem

The \(L_p\) $L_p$ chord Minkowski problem based on chord measures and \(L_p\) $L_p$ 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 \(L_p\) $L_p$ Brunn–Minkowski theory. Xi et al. (Adv Nonlinear Stud 23:20220041, 2023) using variational methods gave a measure solution when \(p > 1\) $p>1$ and \(0<p<1\) $0<p<1$ in the symmetric case. Recently, Guo et al. (Math Ann 2023. 10.1007/s00208-023-02721-8) also obtained a measure solution for \(0\le p<1\) $0\le p<1$ by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the \(L_p\) $L_p$ chord Minkowski problem by replacing p with a fixed continuous function \(\varphi :(0,\infty )\rightarrow (0,\infty )\) $\phi :(0,\infty )\to (0,\infty )$ , and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.