A Geometric Proof of the Colored Tverberg Theorem

作者：Matousek Jiri; Tancer Martin; Wagner Uli^*

来源：DISCRETE %26 COMPUTATIONAL GEOMETRY, 2012, 47(2): 245-265.

DOI：10.1007/s00454-011-9368-2

• Convexity
• Tverberg%26apos
• s theorem
• Colored Tverberg theorem
• Continuous motion

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set CaS,a%26quot;e (d) of cardinality (d+1)t, partitioned into t-point subsets C (1),C (2),aEuro broken vertical bar,C (d+1) (which we think of as color classes; e.g., the points of C (1) are red, the points of C (2) blue, etc.), there exist r disjoint sets R(1), R(2), ... ,R(r) subset of C that are rainbow, meaning that |R (i) boolean AND C(j)| %26lt;= 1 for every i, j, and whose convex hulls all have a common point. %26lt;br%26gt;All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by BlagojeviA double dagger, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.

• 出版日期2012-3