Abstract
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.
Continuous Maps from Spheres Converging to Boundaries of Convex Hulls