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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Matoušek, Lectures on Discrete Geometry, Springer-Verlag, New York 2002.
M. A. Perles and M. Sigron, Strong general position, http://arxiv.org/abs/1409.2899.
M. A. Perles and M. Sigron, A generalization of Tverberg’s Theorem, http://arxiv.org/abs/0710.4668.
J. R. Reay, Several generalizations of Tverberg’s theorem, Israel J. Math. 34 (1979), 238–244.
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.
K. S. Sarkaria, Tverberg’s theorem via number fields, Israel J. Math. 79 (1992), 317–320.
H. Tverberg, A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.
H. Tverberg, A generalization of Radon’s theorem 2, Bull. Austral. Math. Soc. 24 (1981), 321–325.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1434-2