Smoothness and geometry of boundaries associated to skeletal structures I: Sufficient conditions for smoothness

Abstract

We introduce a skeletal structure (M, U) in ℝn+1, which is an n-dimensional Whitney stratified set M on which is defined a multivalued "radial vector field" U. This is an extension of notion of the Blum medial axis of a region in ℝn+1 with generic smooth boundary. For such a skeletal structure there is defined an "associated boundary" B, We introduce geometric invariants of the radial vector field U on M and a "radial flow" from M to B. Together these allow us to provide sufficient numerical conditions for the smoothness of the boundary B as well as allowing us to determine its geometry. In the course of the proof, we establish the existence of a tubular neighborhood for such a Whitney stratified set.