The medial axis of a plane
domain is defined to be the set of the centers of the maximal inscribed disks. It is
essentially the cut loci of the inward unit normal bundle of the boundary. We prove
that if a plane domain has finite number of boundary curves each of which consists of
finite number of real analytic pieces, then the medial axis is a connected geometric
graph in ℝ2 with finitely many vertices and edges. And each edge is a real
analytic curve which can be extended in the C1 manner at the end vertices. We
clarify the relation between the vertex degree and the local geometry of
the domain. We also analyze various continuity and regularity results in
detail, and show that the medial axis is a strong deformation retract of the
domain which means in the practical sense that it retains all the topological
informations of the domain. We also obtain parallel results for the medial axis
transform.
