A Generalisation of Tverberg's Theorem

The well know theorem of Tverberg states that if n ≥ (d+1)(r − 1)+1 then one can partition any set of n points in R to r disjoint subsets whose convex hulls have a common point. The numbers T (d, r) = (d + 1)(r − 1) + 1 are known as Tverberg numbers. Reay asks the following question: if we add an additional parameter k (2 ≤ k ≤ r) what is the minimal number of points we need in order to guarantee that there exists an r partition of them such that any k of the r convex hulls intersect. This minimal number is denoted by T (d, r, k). Reay conjectured that T (d, r, k) = T (d, r) for all d, r and k. In this article we prove that this is true for the following cases: when k ≥ [ d+3 2 ] or when d < rk r−k − 1 and for the specific values d = 3, r = 4, k = 2 and d = 5, r = 3, k = 2.