Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Given n distinct points$\mathbf {x}_1, \ldots , \mathbf {x}_n$ 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, easily computable one-parameter family of continuous maps$\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for$\varepsilon> 0$, are defined on the$(d-1)$-dimensional sphere, and whose images$\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension$1$ submanifolds contained in the interior of K. Moreover, as the parameter$\varepsilon $ goes to$0^+$, the images$\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ 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 these results are included.