有色Tverberg剖分中的等系数和容差
Equal coefficients and tolerance in coloured Tverberg partitions
摘要：
彩色特沃伯格定理是由Bárány，Lovász和Füredi提出的，并询问是否对任何d+1集（被认为是颜色类）的k个点，每个在R^d有一个分区，他们成为k个彩色集的凸包相交。当d= 1，2或k+1是素数时，这是已知的。本文证明了：如果要求每个色类中的凸组合的系数相同，则（k-1）d+1个色类是充要的.我们还检查会发生什么，如果我们希望的凸包的彩色集相交，即使我们删除任何r的颜色类。也就是说，如果我们有（r+1）（k-1）d+1个颜色类，每个颜色类都有k个点，则将它们划分为k个彩色集合，使得它们使用相同的系数相交，而不管去除了哪r个颜色类。我们还调查的情况下k=2和Gale变换的关系，得到一个变化的彩色Radon定理。
英文摘要：
The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d=1,2 or k+1 is prime. In this paper we show that (k-1)d+1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes. Namely, if we have (r+1)(k-1)d+1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed. We also investigate the relation of the case k=2 and the Gale transform, obtaining a variation of the coloured Radon theorem.
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