Colorful Intersections and Tverberg Partitions

Abstract

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 $d\ge m\ge 1$ and $k$ a prime power. Suppose $F_1,F_2,\dots ,F_m$ are families of convex sets in ${\mathbb{R}}^d$, each of size $n>(\frac{d}{m}+1)(k-1)$, such that for any choice $C_i\in F_i$ we have ${\bigcap }_{i=1}^m{C}_i\ne \varnothing$. Then, one of the families $F_i$ admits a Tverberg $k$-partition. That is, one of the $F_i$ 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 ${\mathbb{R}}^d$ that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.

accepted to SOCG2024