Nonoverlap of the star unfolding

The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.