论文标题
凸壳的理性地图和边界
Rational Maps and Boundaries of Convex Hulls
论文作者
论文摘要
如果$ c_n（\ mathbb {r}^d）$表示$ n $不同点的配置空间，则$ \ mathbb {r}^d $，我们构造了一个映射$（f_m），$ $ $ $ $ $ m \ geq 1 $，其中\ [f_m：c_n（c_n（c_n（c_n（c_n（c_n（c_n（c_n）， \ mathbb {r}^d \]是真实的分析性，并且具有C_n（\ Mathbb {r}^d）$的任何$ \ Mathbf {X} \ in任何$ \ Mathbf {x} \ a的属性，任何$ m \ geq 1 $ \ mathbb {r}^d $是一张理性地图，其图像位于$ \ mathbf {x} $的凸壳中。我们的近似猜想是，对于任何$ \ mathbf {x} \在c_n（\ m athbb {r}^d）$中，在我们的映射$ f_m（\ mathbf {x}， - ）$下的球体$ s^{d-1} $的图像是$ nate $ nath $ nath $ naull的近似值。更准确地说，我们猜测\ [\ [\ propatorName {lim} _ {m \ to \ infty} d_h \ left（f_m（\ mathbf {x}， - ）（s^{d-1}）（s^{d-1}），\，\，\ partial \ partial \ partAtorname { $ d_h（ - ， - ）$是hausdorff距离，$ \ operatorname {cons}（\ mathbf {x}）$是$ \ mathbf {x} $ of $ \ mathbf {x} $和$ \ partial $的凸壳。这项工作将介绍计算机生成的图。
If $C_n(\mathbb{R}^d)$ denotes the configuration space of $n$ distinct points in $\mathbb{R}^d$, we construct a sequence of maps $(f_m),$ $m \geq 1$, where \[f_m: C_n(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d\] is real analytic, and has the property that for any $\mathbf{x} \in C_n(\mathbb{R}^d)$ and any $m \geq 1$, the map $f_m(\mathbf{x},-): \mathbb{R}^d \to \mathbb{R}^d$ is a rational map whose image lies in the convex hull of $\mathbf{x}$. Our Approximation Conjecture is that for any $\mathbf{x} \in C_n(\mathbb{R}^d)$, the image of the sphere $S^{d-1}$ under our map $f_m(\mathbf{x},-)$ is an approximation of the boundary of the convex hull of $\mathbf{x}$. More precisely, we conjecture that \[ \operatorname{lim}_{m \to \infty} d_H\left(f_m(\mathbf{x},-)(S^{d-1}), \,\partial \operatorname{Conv}(\mathbf{x}) \right) = 0, \] where $d_H(-,-)$ is the Hausdorff distance, $\operatorname{Conv}(\mathbf{x})$ is the convex hull of $\mathbf{x}$ and $\partial$ is the boundary operator. Computer generated plots will be presented in this work.
