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

TL;DR

This paper improves Erd"os's 1946 theorem by establishing new regularity properties of the ambiguous locus, showing it can be covered by surfaces with convexity and $C^2$ regularity, enhancing understanding of its geometric structure.

Contribution

The paper provides a new regularity result for the surfaces covering the ambiguous locus, emphasizing convexity and $C^2$ smoothness, which was not previously established.

Findings

The ambiguous locus can be covered by countably many convex $C^2$ surfaces.

These surfaces have finite $(n-1)$-dimensional measure.

The regularity results improve the geometric understanding of the medial axis.

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.