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arXiv · 2308.05332

Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

Xia Zhao·Peibiao Zhao
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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 $φ:(0,\infty)\rightarrow(0,\infty)$, and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.

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Xia Zhao, Peibiao Zhao. 2024-04-30. Flow by Gauss Curvature to the orlicz Chord Minkowski Problem. https://arxiv.org/abs/2308.05332

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