Several generalizations of Tverberg’s theorem

@article{Reay1979SeveralGO,
  title={Several generalizations of Tverberg’s theorem},
  author={John R. Reay},
  journal={Israel Journal of Mathematics},
  year={1979},
  volume={34},
  pages={238-244}
}
  • J. R. Reay
  • Published 1 September 1979
  • Mathematics
  • Israel Journal of Mathematics
AbstractIn a generalization of Radon’s theorem, Tverberg showed that each setS of at least (d+1) (r − 1)+1 points inRdhas anr-partition into (pair wise disjoint) subsetsS =S1 ∪ … ∪Srso that $$\bigcap\nolimits_i^r {\underline{\underline {}} } _1 $$ convSi# Ø. This note considers the following more general problems: (1) How large mustS σRdbe to assure thatS has anr-partitionS=S1∪ … ∪Srso that eachn members of the family {convSi∼i-1r have non-empty intersection, where 1<=n<=r. (2) How large mustS… 
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and proved it in the case n = 2. In the case n > 2, Birch proved a weaker result, with r(n+l) — n replaced by rn(n+l) — n — n+ 1. This was, for most r and n, an improvement of the earlier result, by
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