Directional curvature and medial axis

Abstract

The medial axis $M_X$ of a closed set $X\subset {\mathbb{R}}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C^1$-smooth hypersurfaces and extended to any codimension by Białożyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case.