Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Given n distinct points in, let K denote their convex hull, which we assume to be d-dimensional, and its-dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps which, for 0$]]>, are defined on the-dimensional sphere, and whose images are codimension submanifolds contained in the interior of K. Moreover, as the parameter goes to, the images converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.