Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

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 $0\le p<1$ 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 $\phi :(0,\mathrm{\infty })\to (0,\mathrm{\infty })$, and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.