Some variations on Tverberg’s theorem

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 R^d 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 R^d 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.