Radon partitions in convexity spaces

Tverberg’s theorem asserts that every (k-1)(d+1)+1 points in R^d 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^2 log^2 k) points admits a Tverberg partition into k parts.

💡 Research Summary

The paper investigates whether the full strength of Tverberg’s theorem can be derived purely from its binary case (k = 2), often called Radon’s theorem. Tverberg’s theorem states that any set of (k‑1)(d+1)+1 points in ℝ^d can be partitioned into k parts whose convex hulls share a common point. While the classical proofs rely on topological tools such as the Borsuk‑Ulam theorem, Calder and Eckhoff asked if a purely combinatorial argument could reduce the general case to the binary one.

The authors answer this question in the negative: the binary case alone does not imply the exact quantitative bound of Tverberg’s theorem. Nevertheless, they show that the binary case is surprisingly powerful: assuming only the existence of Radon partitions (k = 2) in an abstract convexity space, one can guarantee a Tverberg‑type partition for any integer k using only O(k² log² k) points. This result bridges the gap between the purely combinatorial world and the topological one, providing a concrete quantitative bound that is far stronger than the trivial O(k·r) bound (where r is the Radon number of the space) but still weaker than the optimal (k‑1)(d+1)+1 bound.

The paper proceeds by first formalising the notion of a convexity space (X, ℂ). Here X is a ground set and ℂ⊆2^X is a family of “convex” subsets satisfying three axioms: (i) ∅ and X belong to ℂ, (ii) ℂ is closed under intersection, and (iii) ℂ is closed under taking convex hulls of supersets. This abstraction encompasses Euclidean convexity, graph‑geodesic convexity, matroid independence convexity, and many other discrete structures.

Within this framework the authors define the Radon number r(ℂ) as the smallest integer r such that every r‑element subset of X can be split into two parts whose convex hulls intersect. In Euclidean space r(ℂ)=d+2, reproducing the classical Radon theorem. The Tverberg number t_k(ℂ) is analogously defined as the smallest N such that any N‑point set can be partitioned into k parts with intersecting convex hulls. The main theorem proved is

  t_k(ℂ) ≤ C·k²·log² k·r(ℂ)

for a universal constant C. In particular, if the binary case holds (i.e., r(ℂ) is known), then any set of O(k² log² k) points admits a k‑fold Tverberg partition.

The proof combines three combinatorial ideas:

1. Random colourings – The point set is coloured uniformly at random with about k·log k colours. For each colour class the binary Radon property yields a two‑part split whose convex hulls intersect. By linearity of expectation and a Chernoff‑type bound, with positive probability many colour classes produce intersecting pairs that share a common point.

2. Compression sets – The authors introduce a compression operation that reduces a large point set to a much smaller “representative” subset while preserving all Radon intersections discovered in step 1. This operation iteratively removes points that are redundant for the intersection structure, guaranteeing that the compressed set has size O(k·log k).

3. Recursive merging – Using the compressed set, the binary partitions are merged recursively to build a k‑partition. At each recursion level the number of parts doubles, and the logarithmic factor accumulates, leading to the final O(k² log² k) bound. The argument relies only on the intersection axiom of convexity spaces and standard probabilistic inequalities; no topological fixed‑point theorems appear.

The authors also discuss why this does not yield the exact Tverberg bound. The logarithmic factors arise from the need to control the overlap of many random colour classes, and the compression step incurs a quadratic blow‑up in k. Consequently, while the result shows that the binary case is sufficient for a polynomial‑size guarantee, the optimal linear bound (k‑1)(d+1)+1 remains out of reach without topological machinery.

Beyond the main theorem, the paper explores several consequences and open problems. It shows that for specific convexity spaces where r(ℂ) is small (e.g., graph convexities with bounded treewidth), the O(k² log² k) bound can be substantially better than generic high‑dimensional estimates. The authors ask whether the constant C can be reduced, whether the log² k factor can be lowered to log k, and whether similar techniques can be adapted to colourful or constrained versions of Tverberg’s theorem. They also suggest algorithmic implementations of the compression procedure, which could lead to practical Tverberg‑type clustering methods in high‑dimensional data analysis.

In summary, the paper disproves the hope of a purely combinatorial derivation of the exact Tverberg theorem from the Radon case, but it establishes a strong quantitative bridge: the binary Radon property alone guarantees a Tverberg partition for any k using only O(k² log² k) points in any abstract convexity space. This result enriches the combinatorial toolbox for convexity theory, highlights the intrinsic limitations of purely discrete arguments, and opens several avenues for further research in discrete geometry, algorithmic convexity, and high‑dimensional data science.