A Generalisation of Tverberg's Theorem

Abstract

We will prove the following generalisation of Tverberg's Theorem: given a set S⊂ℝ d of (r+1)(k-1)(d+1)+1 points, there is a partition of S in k sets A 1,A 2,..,A k such that for any C⊂S of at most r points, the convex hulls of A 1\C,A 2\C,..,A k\C are intersecting. This was conjectured first by Natalia García-Colín (Ph. D. thesis, University College of London, 2007). © 2012 Springer Science+Business Media, LLC.