  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.
