Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Abstract

Given $n$ distinct points ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_n$ in ${\mathbb{R}}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B=\partial K$ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps ${\mathbf{f}}_{\epsilon }:{\mathbb{S}}^{d-1}\to K$ which, for $\epsilon >0$, are defined on the $(d-1)$-dimensional sphere and have the property that the images ${\mathbf{f}}_{\epsilon }({\mathbb{S}}^{d-1})$ are codimension $1$ submanifolds contained in the interior of $K$. Moreover, as the parameter $\epsilon$ goes to $0^{+}$, the images ${\mathbf{f}}_{\epsilon }({\mathbb{S}}^{d-1})$ 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 our results will be presented.

26 pages