Abstract The medial axis transform is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning, and tolerance specification. In this paper, the theory of the medial axis transform for 3-D objects is developed. For objects with piecewise C 2 boundaries, relationships between the curvature of the boundary and the position of the medial axis are developed. For n -dimensional submanifolds of R n with boundaries which are piecewise C 2 and completely G 1 , a deformation retract is set up between each object and its medial axis, which demonstrates that if the object is path connected, then so is its medial axis. Finally, it is proven that path connected polyhedral solids without cavities have path connected medial axes.