Equal coefficients and tolerance in coloured Tverberg partitions

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• 作者：Pablo Soberón
• 关键词：52A35 ; 52A37 ; 05A18
• 刊名：Combinatorica
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：35
• 期：2
• 页码：235-252
• 全文大小：437 KB
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• 作者单位：Pablo Soberón (1)
  
  1. Department of Mathematics, University College London, Gower Sreet, London, WC1E 6BT, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1439-6912

文摘

The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi [4] and asks whether for any d+1 sets (considered as colour classes) of k points each in ?sup> d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d=1;2 [5] or k+1 is prime [7]. In this paper we show that (k?)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. This result is actually a generalisation of Tverberg’s classic theorem on the intersection of convex hulls [27]. 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, and its relation to other colourful variants of Tverberg’s theorem. We investigate the relation of the case k=2 and the Gale transform, obtaining a variation of the coloured Radon theorem. We then show applications of these results to purely combinatorial problems.