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Regular Polygonal Partitions of a Tverberg Type
Discrete & Computational Geometry ( IF 0.6 ) Pub Date : 2021-04-01 , DOI: 10.1007/s00454-021-00288-2
Leah Leiner , Steven Simon

A seminal theorem of Tverberg states that any set of \(T(r,d)=(r-1)(d+1)+1\) points in \({\mathbb {R}}^{d}\) can be partitioned into r subsets whose convex hulls have non-empty r-fold intersection. Almost any collection of fewer points in \({\mathbb {R}}^{d}\) cannot be so divided, and in these cases we ask if the set can nonetheless be P(rd)-partitioned, i.e., split into r subsets so that there exist r points, one from each resulting convex hull, which form the vertex set of a prescribed convex d-polytope P(rd). Our main theorem shows that this is the case for any generic \(T(r,2)-2\) points in the plane and any \(r\ge 3\) when \(P(r,2)=P_{r}\) is a regular r-gon, and moreover that \(T(r,2)-2\) is tight. For higher dimensional polytopes and \(r=r_{1}\cdots r_{k}\), \(r_{i} \ge 3\), this generalizes to \(T(r,2k)-2k\) generic points in \({\mathbb {R}}^{2k}\) and orthogonal products \(P(r,2k)=P_{r_{1}}\times \cdots \times P_{r_{k}}\) of regular polygons, and likewise to \(T(2r,2k+1)-(2k+1)\) points in \({\mathbb {R}}^{2k+1}\) and the product polytopes \(P(2r,2k+1)=P_{r_{1}}\times \cdots \times P_{r_{k}}\times P_{2}\). As with Tverberg’s original theorem, our results admit topological generalizations when r is a prime power, and, using the “constraint method” of Blagojević, Frick, and Ziegler, allow for dimensionally restricted versions of a van Kampen–Flores type and colored analogues in the fashion of Soberón.



中文翻译:

Tverberg类型的规则多边形分区

特维尔贝格(Tverberg)的一个开明定理指出,\(T(r,d)=(r-1)(d + 1)+1 \)的任何集合都指向\({\ mathbb {R}} ^ {d} \)可以划分为r个子集,这些子集的凸包具有非空的r折交集。\({\ mathbb {R}} ^ {d} \)中几乎所有较少点的集合都不能如此划分,在这种情况下,我们询问集合是否仍可以进行Pr,  d)划分,即分成r个子集,以便存在r个点(每个所得凸包中的一个),这些点形成指定的凸d-多面体Pr,  d)。我们的主要定理表明这是任何通用的情况下\(T(R,2)-2 \)在平面和任何点\(R \ GE 3 \)\(P(R,2)= P_ { r} \)是一个正则r -gon,而且\(T(r,2)-2 \)是紧的。对于高维多曲面\(r = r_ {1} \ cdots r_ {k} \)\(r_ {i} \ ge 3 \),这一般化为\(T(r,2k)-2k \)泛型在点\({\ mathbb {R}} ^ {2K} \)和正交产品P_ {R_ {1}} \倍\ cdots \倍\(P(R,2K)= P_ {R_ {K}} \ )的规则多边形,同样以\(T(2r,2k + 1)-(2k + 1)\)\({{mathbb {R}} ^ {2k + 1} \)和乘积多峰\(P(2r,2k + 1)= P_ {r_ {1}} \ times \ cdots \ times P_ {r_ {k }} \ P_ {2} \)。与特维尔伯格的原始定理一样,当r为素数幂时,我们的结果也可以接受拓扑概括,并且使用Blagojević,Frick和Ziegler的“约束方法”,可以对van Kampen–Flores类型和彩色类似物进行尺寸限制Soberón的时尚。

更新日期:2021-04-01
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