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

Let P be a convex polyhedron in R3. 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 the unit-sphere, the skeleton is a (0.999 · π)-spanner. If the vertices are very close to this sphere, then the skeleton is not necessarily a spanner. For the case when the boundary of P is between two concentric spheres of radii 1 and R > 1, 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 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.