Equal coefficients and tolerance in coloured Tverberg partitions

Abstract

The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi [4] and asks whether for any d+1 sets (considered as colour classes) of k points each in ℝ^d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d=1;2 [5] or k+1 is prime [7]. In this paper we show that (k−1)d+1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. This result is actually a generalisation of Tverberg’s classic theorem on the intersection of convex hulls [27]. We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes, and its relation to other colourful variants of Tverberg’s theorem. We investigate the relation of the case k=2 and the Gale transform, obtaining a variation of the coloured Radon theorem. We then show applications of these results to purely combinatorial problems.