  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.
