Tverberg plus constraints

Abstract

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this, we introduce a proof technique that combines a concept of ‘Tverberg unavoidable subcomplexes’ with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus, we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Živaljević and Vrećica (‘The colored Tverberg's problem and complexes of injective functions’, J. Combin. Theory, Ser. A 61 (1992) 309–318). We also get a new strengthened version of the generalized van Kampen–Flores theorem by Sarkaria (‘A generalized van Kampen–Flores theorem’, Proc. Amer. Math. Soc. 11 (1991) 559–565) and Volovikov (‘On the van Kampen–Flores theorem’, Math. Notes (5) 59 (1996) 477–481), an affine version of their ‘j‐wise disjoint’ Tverberg theorem, and a topological version of Soberón's result (‘Equal coefficients and tolerance in coloured Tverberg partitions’, Proceeding of the 29th Annual Symposium on Computational Geometry (SoCG) (ACM, Rio de Janeiro, 2013), 91–96) on Tverberg points with equal barycentric coordinates.

Tverberg’s theorem is 50 years old: A survey

“…The proof, rediscovered by Blagojević, Frick, and Ziegler [BFZ14] is almost identical to the previous proof for the case r = 2. It has two ingredients: one is the topological Tverberg theorem, the other is the constraint method (or the pigeonhole principle).…”

“…If we have fewer than d + 1 color classes in the colorful Tverberg theorem, we cannot guarantee the existence of a colorful Tverberg partition into r parts for sufficiently large r. This follows simply because colorful simplices have positive co-dimension. However, when the co-dimension is not a problem, a similar proof to the one above yields the following result and its natural topological version [VŽ94,BFZ14].…”

“…This gives us a partition into r + 1 parts, and we can simply drop the part containing p 0 and redistribute its points to have a partition as in Conjecture 3.1. The constraint method by Blagojević, Ziegler, and Frick [BFZ14], mentioned in Section 2 also gives colorful Tverberg's results. We showcase here how it implies the bound t(d, r) ≤ 2r − 1 when r is a prime power.…”

Tverberg's Theorem at 50: Extensions and Counterexamples

“…So it came as quite a surprise that there is a really simple "constraint method" to get virtually all of these extensions directly from the original topological Tverberg theorem. This observation arose in our collaboration [2] with Florian Frick at a blackboard at Arnimallee 2, a villa that is part of the Mathematical Institute of FU Berlin. We couldn't believe that it was so easy!…”

“…First, with Florian Frick [2] we designed a "constraint method" that yields colored versions from the original "topological Tverberg theorem" quite easily. Second, Isaac Mabillard and Uli Wagner in Vienna developed an " -fold Whitney trick," and Florian Frick in Berlin noticed that combined with the constraint method this yields counterexamples for all ≥ 6 that are not prime powers.…”