Delta-convex structure of the singular set of distance functions

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics Pub Date : 2024-03-07 DOI:10.1002/cpa.22195

Tatsuya Miura, Minoru Tanaka

引用次数: 0

Abstract

For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta-convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

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距离函数奇异集的三角凸结构

对于从任何完整芬斯勒流形的任何封闭子集出发的距离函数，我们证明奇异集等于△凸超曲面的可数联合，直到一个标度为二的例外集。此外，在维数二中，整个奇异集等于直到孤立点的△凸约旦弧的可数联盟。即使在标准欧几里得空间中，这些结果也是新的，而且从正则性的角度来看，这些结果是最优的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.