We prove a definable/subanalytic version of a useful lemma, presumably due to John Nash, concerning the points realizing the Euclidean distance to an analytic submanifold of R n .We present a parameter version of the main result and we discuss the properties of the multifunction obtained.

This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set X ⊂ R n , which are reached by the medial axis.

No abstract