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arXiv · 2011.14508

On an old theorem of Erd\"os about ambiguous locus

Piotr Hajłasz
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Abstract

Erd\"os proved in 1946 that if a set $E\subset\mathbb{R}^n$ is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in $\mathbb{R}^n$ with the property that the nearest point in $E$ is not unique, can be covered by countably many surfaces, each of finite $(n-1)$-dimensional measure. We improve the result by obtaining a new regularity result for these surfaces in terms of convexity and $C^2$ regularity.

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Piotr Hajłasz. 2020-11-30. On an old theorem of Erd\"os about ambiguous locus. https://arxiv.org/abs/2011.14508

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