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 Rᵈ, each of size n > (d/m+1)(k-1), such that for any choice Cᵢ∈ Fᵢ we have ᵢ₌₁ᵐCᵢ≠ ∅. 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 Rᵈ that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.