The tangent cone, the dimension and the frontier of a medial axis

Abstract: This paper aims to establish a relation between the tangent cone of the medial axis of X at a given point a of R^n$ and the medial axis of the set of points in X realising the distance d(a,X). As a consequence, a lower bound for the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a) is obtained. This appears to be the missing link to the full description of the medial axis' dimension. Further study of potentially troublesome points on the frontier of the medial axis is also provided, resulting in their characterisation in terms of the reaching radius.