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 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 have the property that the 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 our results will be presented.