When the medial axis meets the singularities

. In this survey we present recent results in the study of the medial axes of sets definable in polynomially bounded o-minimal structures. We take the novel point of view of singularity theory. Indeed, it has been observed only recently that the medial axis — i.e. the set of points with more than one closest point to a given closed set X ⊂ R n (with respect to the Euclidean distance) — reaches some singular points of X bringing along some metric information about them.