Radon partitions in convexity spaces

Tverberg’s theorem asserts that every (k − 1)(d + 1) + 1 points in R can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction of Tverberg’s theorem from the special case k = 2. We dash the hopes of a purely combinatorial deduction, but show that the case k = 2 does imply that every set of O(k log k) points admits a Tverberg partition into k parts. Introduction Radon’s lemma [Rad21] states that every set P of d + 2 points in Rd can be partitioned into two classes P = P1 ∪ P2 so that the convex hulls of P1 and P2 intersect. Birch [Bir59] (for d = 2) and Tverberg [Tve66] (for general d) extended Radon’s theorem to the analogous statement for partitions of a set into more than two parts: For a set P ⊂ Rd of |P | ≥ (k−1)(d+1)+1 points there is a partition P = P1∪ · · ·∪Pk into k parts, such that the intersection of the convex hulls conv P1 ∩ · · · ∩ convPk is non-empty. The bound of (k−1)(d+1)+1 is sharp, as witnessed by any set of points in sufficiently general position. Calder [Cal71] conjectured and Eckhoff [Eck79] speculated that Tverberg’s theorem is a consequence of Radon’s theorem in the context of abstract convexity spaces. The conjecture, which we now present, is commonly referred as “Eckhoff’s conjecture”, and we will maintain this tradition to avoid additional confusion. If true, the conjecture would have provided a purely combinatorial proof of Tverberg’s theorem. However, we will show that the conjecture is false. A convexity space on the ground set X is a family F ⊂ 2X of subsets of X, called convex sets, such as both ∅ and X are convex, and intersection of any collection of convex sets is convex. For example, the familiar convex sets in Rd form a convexity space on R d. Among the other examples are axis-parallel boxes in Rd, finite subsets on any ground set, closed sets in any topological space (see the book [vdV93] for a through overview of convexity spaces). If the ground set X in the convexity space (X,F) is clear from the The paper is in public domain, and is not protected by copyright. [email protected]. Centre for Mathematical Sciences, Cambridge CB3 0WB, England and Churchill College, Cambridge CB3 0DS, England.