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.