通过Tverberg划分和Gale对偶获得扇形分布
Fan distributions via Tverberg partitions and Gale duality
AI总结：
通过Tverberg划分和Gale对偶，研究有限点集在$\mathbb{R}^d$中的等分布问题，证明在特定条件下存在满足点分布要求的$r$-扇形结构。
摘要：
等分理论，从经典的汉尼拔定理开始，旨在通过预设的几何分割确定全维区域来公平分配有限点集。在这里，我们研究通过预设的低维子集对有限点集在$\mathbb{R}^d$中的等分布。我们的主要结果表明，如果$r \geq 3$是一个素数幂，则对于任意一个足够小的点集$X$在$\mathbb{R}^d$中的$m$-染色，存在一个$r$-扇形，即由$r$个“半平面”组成，这些半平面的余维数为$r-2$，并围绕一个共同的$r-1$余维的仿射子空间中心。以这种方式捕捉所有点，使得每个半平面中每个颜色类的点数不超过总点数的$r$-分之一。在$\mathbb{R}^d$中所需的点数在$m \geq 2$时基本上是紧致的。此外，我们以类似Dolnikov对汉尼拔定理的超平面穿越推广的方式扩展我们的等分布结果到“穿刺”分布。通过类比最近Frick等人工作的模式，我们的结果通过将Gale对偶应用于拓扑Tverberg型定理的线性情况来获得。最后，我们通过建立Sarkaria拓扑Tverberg型定理的多重交集版本，将我们的分布结果扩展到多个$r$-扇形。
英文摘要：
Equipartition theory, beginning with the classical ham sandwich theorem, seeks the fair division of finite point sets in $\mathbb{R}^d$ by the full-dimensional regions determined by a prescribed geometric dissection of $\mathbb{R}^d$. Here we examine $\textit{equidistributions}$ of finite point sets in $\mathbb{R}^d$ by prescribed $\textit{low dimensional}$ subsets. Our main result states that if $r\geq 3$ is a prime power, then for any $m$-coloring of a sufficiently small point set $X$ in $\mathbb{R}^d$, there exists an $r$-fan in $\mathbb{R}^d$ -- that is, the union of $r$ ``half-flats'' of codimension $r-2$ centered about a common $(r-1)$-codimensional affine subspace -- which captures all the points of $X$ in such a way that each half-flat contains at most an $r$-th of the points from each color class. The number of points in $\mathbb{R}^d$ we require for this is essentially tight when $m\geq 2$. Additionally, we extend our equidistribution results to ''piercing'' distributions in a similar fashion to Dolnikov's hyperplane transversal generalization of the ham sandwich theorem. By analogy with recent work of Frick et al., our results are obtained by applying Gale duality to linear cases of topological Tverberg-type theorems. Finally, we extend our distribution results to multiple $r$-fans after establishing a multiple intersection version of a topological Tverberg-type theorem due to Sarkaria.
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