Equal coefficients and tolerance in coloured tverberg partitions

Abstract

The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi [2] 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 [3] or k + 1 is prime [5]. 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. 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. Namely, if we have (r + 1)(k − 1)d + 1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed. We also investigate the relation of the case k = 2 and the Gale transform. We then show applications of these results to purely combinatorial problems.