Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

TL;DR

The paper constructs a family of continuous maps from spheres into convex bodies in Euclidean space, which converge to the boundary of the convex hull as a parameter approaches zero, revealing geometric and topological properties.

Contribution

It introduces explicit maps from spheres into convex bodies that converge to the boundary, linking convex geometry with set-valued homology and the Gauss map.

Findings

Maps are continuous and their images are codimension 1 submanifolds inside the convex body.

As the parameter approaches zero, the images converge to the boundary of the convex hull.

The approach connects convex geometry, topology, and the Gauss map of polytopes.

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 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.