Regularity of the set of points with a non-unique metric projection

Piotr Hajłasz

Published 2020-11-30Version 1

Erd\"os proved in 1946 that if a set $E\subset\mathbb{R}^n$ is closed and non-empty, then the set 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 new regularity results for these surfaces in terms of convexity and $C^2$ regularity.