# 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} }

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…

## 25 Citations

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The main results of this paper demonstrate that Tverberg's theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve complexes, as soon as the number of points is large enough.

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The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d…

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This chapter discusses applications and generalizations of the classical theorems of Helly, Radon, and Caratheodory, as well as their ramifications in the context of combinatorial convexity theory.…

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Eckhoff's conjecture for the Τ-Radon numbers r(Τ) of a convexity space. (X,C) says r(Τ) ≦ (r−1)(Τ−1)+1, with r = r(2). The main result of this note is that Eckhoff's conjecture is true in case ¦X¦ ≦…

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Borders are given for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.

A Tverberg-type generalization of the Helly number of a convexity space

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In 1966 H. Tverberg gave a far reaching generalization of the well-known classical theorem of J. Radon. In this paper a similar generalization of the classical Helly theorem is given and it is shown…

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A counterexample for n =9, d =3, m =3 is given to prove the conjecture that conv D m ( S)= C m (S) is known to be true for d =2.

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