On the Number of Birch Partitions

Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness and a lower bound. This implies the first nontrivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments and do not use the equivariant method from topological combinatorics.

[1]  Rade T. Zivaljevic,et al.  Note on a conjecture of sierksma , 1993, Discret. Comput. Geom..

[2]  Murad Ozaydin,et al.  Equivariant Maps for the Symmetric Group , 1987 .

[3]  K. S. Sarkaria.  Tverberg partitions and Borsuk–Ulam theorems , 2000 .

[4]  Jirí Matousek,et al.  Quadratically Many Colorful Simplices , 2007, SIAM J. Discret. Math..

[5]  Tamon Stephen,et al.  Colourful Simplicial Depth , 2006, Discret. Comput. Geom..

[6]  Imre Bárány,et al.  On a Topological Generalization of a Theorem of Tverberg , 1981 .

[7]  Stephan Hell.  Tverberg-type theorems and the Fractional Helly property , 2006 .

[8]  J. Pach,et al.  The number of simplices embracing the origin , 2003 .

[9]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[10]  Helge Tverberg,et al.  On Generalizations of Radon's Theorem and the Ham Sandwich Theorem , 1993, Eur. J. Comb..

[11]  Stephan Hell.  On the number of Tverberg partitions in the prime power case , 2007, Eur. J. Comb..

[12]  J. Matousek,et al.  Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry , 2007 .

[13]  Imre Ba´ra´ny,et al.  Quadratically Many Colorful Simplices , 2007 .

[14]  G. Ziegler.  Lectures on Polytopes , 1994 .

[15]  Stephan Hell,et al.  Tverberg's theorem with constraints , 2007, J. Comb. Theory, Ser. A.

[16]  B. J. Birch,et al.  On 3N points in a plane , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

[18]  A. Volovikov,et al.  On a topological generalization of the Tverberg theorem , 1996 .