  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.
