On an old theorem of Erdős about ambiguous locus
Volume 168 / 2022
Colloquium Mathematicum 168 (2022), 249-256
MSC: 26B25, 28A75, 49J52.
DOI: 10.4064/cm8460-9-2021
Published online: 3 January 2022
Abstract
Erdős 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.
