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.

中文翻译：

---

从会聚到凸壳边界的球体的连续映射

给定n不同点 $\mathbf {x}_1, \ldots , \mathbf {x}_n$ 在 $\mathbb {R}^d$ ， 让ķ表示它们的凸包，我们假设它是d-维数，和 $B = \偏K $ 它的 $(d-1)$ 维边界。我们构建了一个明确的、易于计算的单参数连续映射族 $\mathbf {f}_{\varepsilon } \冒号 \mathbb {S}^{d-1} \to K$ 其中，对于 $\伐普西隆> 0$ , 定义在 $(d-1)$ 维球体，其图像 $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ 是维数 $1$ 包含在内部的子流形ķ. 此外，作为参数 $\伐普西隆$ 去 $0^+$ ， 图像 $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ 作为集合收敛到边界乙的凸包。我们使用来自（球形）多面体的凸几何和集值同调的技术来证明这个定理。我们进一步与多面体的高斯图建立了有趣的关系乙，适当定义。包括几个说明这些结果的计算机图。