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Some variations on Tverberg’s theorem

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Abstract

Define T(d, r) = (d + 1)(r - 1) + 1. A well known theorem of Tverberg states that if n ≥ T(d, r), then one can partition any set of n points in Rd into r pairwise disjoint subsets whose convex hulls have a common point. The numbers T(d, r) are known as Tverberg numbers. Reay added another parameter k (2 ≤ k ≤ r) and asked: what is the smallest number n, such that every set of n points in Rd admits an r-partition, in such a way that each k of the convex hulls of the r parts meet. Call this number T(d, r, k). Reay conjectured that T(d, r, k) = T(d, r) for all d, r and k. In this paper we prove Reay’s conjecture in the following cases: when k ≥ [d+3/2], and also when d < rk/r-k - 1. The conjecture also holds for the specific values d = 3, r = 4, k = 2 and d = 5, r = 3, k = 2.

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References

  1. J. Matoušek, Lectures on Discrete Geometry, Springer-Verlag, New York 2002.

    Book  MATH  Google Scholar 

  2. M. A. Perles and M. Sigron, Strong general position, http://arxiv.org/abs/1409.2899.

  3. M. A. Perles and M. Sigron, A generalization of Tverberg’s Theorem, http://arxiv.org/abs/0710.4668.

  4. J. R. Reay, Several generalizations of Tverberg’s theorem, Israel J. Math. 34 (1979), 238–244.

    Article  MathSciNet  MATH  Google Scholar 

  5. J.-P. Roudneff, Partitions of points into simplices with k-dimensional intersection. Part I: The conic Tverberg’s theorem, Europ. J. Combinatorics 22 (2001), 733–743.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. S. Sarkaria, Tverberg’s theorem via number fields, Israel J. Math. 79 (1992), 317–320.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Tverberg, A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Tverberg, A generalization of Radon’s theorem 2, Bull. Austral. Math. Soc. 24 (1981), 321–325.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel

    Micha A. Perles & Moriah Sigron

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  1. Micha A. Perles
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  2. Moriah Sigron
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Correspondence to Micha A. Perles.

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Perles, M.A., Sigron, M. Some variations on Tverberg’s theorem. Isr. J. Math. 216, 957–972 (2016). https://doi.org/10.1007/s11856-016-1434-2

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  • DOI: https://doi.org/10.1007/s11856-016-1434-2

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