In a 1988 Computational Geometry paper, Bárány, Füredi & Lovász noted that they needed a colored version of Tverberg's theorem. Bárány & Larman proved such a theorem for 3N colored points in a plane, and conjectured a version for d dimensions. A remarkable 1992 paper by Zivaljevic & Vrecica obtained such a result, though not with a tight bound on the number of points.
We now propose a new colored Tverberg theorem, which is tight, which generalizes Tverberg's original theorem - and which has three quite different proofs. Pick your favourite!
(Joint work with Pavle V. Blagojevic and Benjamin Matschke)
- Birch, B. J. : On 3N points in a plane. Proc. Cambridge Philos. Soc. 55 (1959), p.289-293.
Tverberg, H. : A generalization of Radon's theorem. J. London Math. Soc. 41 (1966), p.123-128.
- Bárány, I.; Füredi, Z.; Lovász, L. : On the number of halving planes. Combinatorica 10 (1990), no. 2, p. 175-183.
- Bárány, I.; Larman, D. G. : A colored version of Tverberg's theorem. J. London Math. Soc. (2) 45 (1992), no. 2, p.314-320.
- Živaljević, Rade T.; Vrećica, Siniša T. : The colored Tverberg's problem and complexes of injective functions. J. Combin. Theory Ser. A 61 (1992), no. 2, p.309-318
- Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler : Optimal bounds for the colored Tverberg problem. arXiv