On an old theorem of Erdös about ambiguous locus

Abstract

Erdös 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.