Colorful Intersections and Tverberg Partitions

The colorful Helly theorem and Tverberg’s theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers d ≥ m ≥ 1 and k a prime power. Suppose F1, F2, . . ., Fm are families of convex sets in R^d, each of size n > (_m^d + 1)(k − 1), such that for any choice Ci ∈ Fi we have ^Tm_i=1 Ci ≠ ∅. Then, one of the families Fi admits a Tverberg k-partition. That is, one of the Fi can be partitioned into k nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning r-dimensional transversals to families of convex sets in R^d that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.