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.

中文翻译：

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奇异距离函数集的Delta-凸结构

对于任何完整芬斯勒流形的任何闭子集的距离函数，我们证明奇异集等于三角凸超曲面的可数并集，直到余维二的特殊集合。此外，在二维中，整个奇异集等于 Delta-凸 Jordan 弧到孤立点的可数并集。即使在标准欧几里得空间中，这些结果也是新的，并且从规律性角度来看，这些结果是最优的。