Equal coefficients and tolerance in coloured Tverberg partitions

TL;DR

This paper establishes the minimal number of colour classes needed for coloured Tverberg partitions with equal coefficients, explores intersection robustness under removal of classes, and relates to the coloured Radon theorem.

Contribution

It proves the exact number of colour classes required for equal-coefficient Tverberg partitions and examines intersection stability when classes are removed.

Findings

(k-1)d+1 colour classes are necessary and sufficient for equal coefficients.

Partition exists with intersection even after removing r classes.

Connection to the coloured Radon theorem via Gale transform.

Abstract

The coloured Tverberg theorem was conjectured by B\'ar\'any, Lov\'{a}sz and F\"uredi 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 or k+1 is prime. 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, obtaining a variation of the coloured Radon theorem.