Combinatorial Theory

We prove an extension of Tverberg's classical result on partitioning a point set in \(\mathbb{R}^d\) by replacing the point set with families of convex sets which satisfy the colorful Helly hypothesis. In particular, we show the following for any integers \(d \geq m \geq 1\) and \(r\) a prime power. Suppose \(F_1, F_2, \dots, F_m\) are families of convex sets in \(\mathbb{R}^d\), each of size \({n › (\frac{d}{m}+1)(r-1)}\), such that for every choice \(C_1\in F_1, C_2\in F_2, \dots,C_m \in F_m\) we have \(\bigcap_{i=1}^mC_i\neq \varnothing\). Then, one of the families \(F_i\) admits a Tverberg \(r\)-partition. That is, one of the families \(F_i\) can be partitioned into \(r\) nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we extend the work of Karasev and Montejano concerning geometric transversals to families of convex sets in \(\mathbb{R}^d\) that satisfy the colorful Helly hypothesis.

Mathematics Subject Classifications: 52A35, 57Q70

Keywords: Tverberg's theorem, geometric transversals, topological combinatorics, configuration space/test map, discrete Morse theory