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

This paper establishes a relation between the tangent cone of the medial axis of X at a given point a is an element of R-n and the medial axis of the set of points m(a) in X realising the Euclidean 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 formula appears to be the missing link to the full description of the medial axis' dimension. An extended study of potentially troublesome points on the frontier of the medial axis is also provided, resulting in their characterisation by the recently introduced by Birbrair and Denkowski reaching radius whose definition we simplify.