We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

中文翻译：

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二维欧几里得和黎曼空间中距离函数的临界值集的小

我们研究距离函数的临界值集有多小，该集来自平面中或连接的完整二维黎曼流形中的紧凑（闭）集。我们表明，对于紧凑集，临界值集是紧凑的且 Lebesgue null（这是一个已知结果），并且它具有“局部”（远离 0）间隙长度平方根的有界总和（补充）。在平面情况下，这些局部小的条件被证明是最佳的。这些结果改进并推广了 Fu (1985) 和我们 2012 年早期论文的结果。我们还找到了平面紧集的整个临界值集很小的最佳条件。