Paper · Apr 27, 2023

Smooth solutions to the chord log-Minkowski problem

Jinrong Hu, Yong Huang, Jian Lu

math.APWider flowsarXiv:2304.14220v7
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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 Monge-Ampère 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 Monge-Ampère equation and a nonlocal Gauss curvature flow equation.

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Submitted
Apr 27, 2023
Last revised
Oct 15, 2026 (v7)
Subjects
math.AP

Comments: This version corrects the previous version and updates the references

Other curvature flowsGauss curvature flow

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Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974)