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 sphere have property that images $\mathbf{f}_{\varepsilon}(\mathbb{S}^{d-1})$ codimension $1$ submanifolds contained interior $K$. Moreover, as parameter $\varepsilon$ goes $0^+$, converge, sets, boundary $B$ hull. prove this theorem using techniques from geometry (spherical) polytopes set-valued homology. further establish interesting relationship with Gauss map polytope $B$, appropriately defined. Several computer plots illustrating our results will presented.