Optimal bounds for a colorful Tverberg--Vrecica type problem
By Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler. 2009
59 citations recorded by the source corpus.
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).
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5e2a7b2d81f4c39ddfb1d3e24234e19f59d0270a
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Theory and Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Optimal bounds for the colored Tverberg problem (2009, background): “2 in [7] as a special case of a Tverberg–Vrećica type transversal theorem, accompanied by much more complete cohomological index calculations, which also yield a second new proof that establishes Theorem 2.”