On the Stretch Factor of Convex Polyhedra whose Vertices are (Almost) on a Sphere

Abstract

Let P be a convex simplicial polyhedron in R 3 . The skeleton of P is the graph whose vertices and edges are the vertices and edges of P , respectively. We prove that, if these vertices are on a sphere, the skeleton is a (0.999 • π)-spanner. If the vertices are very close to a sphere, then the skeleton is not necessarily a spanner. For the case when the boundary of P is between two concentric spheres of radii r and R, where R > r > 0, and the angles in all faces are at least θ, we prove that the skeleton is a t-spanner, where t depends only on R/r and θ. One of the ingredients in the proof is a tight upper bound on the geometric dilation of a convex cycle that is contained in an annulus.