Smooth solutions to the chord log-Minkowski problem

Jinrong Hu; Yong Huang; Jian Lu

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arXiv · 2304.14220

Smooth solutions to the chord log-Minkowski problem

Jinrong Hu·Yong Huang·Jian Lu

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Abstract

In integral geometry generalized with Aleksandrov's variational theory, Lutwak-Xi-Yang-Zhang [Comm. Pure Appl. Math. 77 (2024)] recently opened the door to researching the cone-chord measures and their log-Minkowski problem stemming from the chord integrals, named as the chord log-Minkowski problem. In the smooth category, the solvability of the chord log-Minkowski problem amounts to dealing with a nonlocal {M}onge-{A}mp\`ere equation involving a Riesz potential. In this paper, to study the chord log-Minkowski problem, we first present some novel results on the boundary regularity of the Riesz potential. Based on these results, we obtain the regularity and existence for the chord log-Minkowski problem from the perspective of a nonlocal {M}onge-{A}mp\`ere equation and a nonlocal Gauss curvature flow equation.

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Jinrong Hu, Yong Huang, Jian Lu. 2023-04-27. Smooth solutions to the chord log-Minkowski problem. https://arxiv.org/abs/2304.14220

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