The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C⊂ℝd of cardinality (d+1)t, partitioned into t-point subsets C1,C2,…,Cd+1 (which we think of as color classes; e.g., the points of C1 are red, the points of C2 blue, etc.), there exist r disjoint sets R1,R2,…,Rr⊆C that are rainbow, meaning that |Ri∩Cj|≤1 for every i,j, and whose convex hulls all have a common point.
All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.