论文标题
连续地图从汇聚到凸面边界的球体
Continuous Maps from Spheres Converging to Boundaries of Convex Hulls
论文作者
论文摘要
给定$ n $不同的点$ \ mathbf {x} _1，\ ldots，\ mathbf {x} _n $ in $ \ mathbb {r}^d $，让$ k $表示他们的凸赫尔，我们假定为$ d $ d $ d $ -dimensional，$ b = p partial k $ its $ its $ its $（$ its $ undials $（dimention）$（d-dimence dimence nource（d-dimence）。 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}）$ as codimension $ 1 $ submanifolds包含在$ k $的内部。此外，随着参数$ \ varepsilon $转至$ 0^+$，图像$ \ mathbf {f} _ {\ varepsilon}（\ Mathbb {s}}^{d-1}）$作为集合的convex hull的边界$ b $。我们使用（球形）多型和集值同源性的凸几何形状的技术证明了这种定理。我们进一步与适当定义的polytope $ b $的高斯地图建立了有趣的关系。将介绍一些说明我们结果的计算机图。
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.
