Optimal bounds for a colorful Tverberg--Vrecica type problem

TL;DR

This paper establishes optimal bounds for a colorful Tverberg-Vrecica type problem involving partitions of colored point collections in Euclidean space, with new proofs and topological results.

Contribution

It provides the first optimal bounds for the problem, introduces two alternative proofs for a special case, and develops new topological tools including a Borsuk-Ulam type theorem.

Findings

Optimal bounds for the colorful Tverberg-Vrecica problem.

Two alternative proofs for the case k=0.

A new Borsuk-Ulam type theorem for (Z_p)^m-equivariant bundles.

Abstract

We prove the following optimal colorful Tverberg-Vrecica type transversal theorem: For prime r and for any k+1 colored collections of points C^l of size |C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k) is even or k=0. Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0 (our optimal colored Tverberg theorem (2009)), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk-Ulam type theorem for (Z_p)^m-equivariant bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).