Partitions of Points into Simplices withk-dimensional Intersection. Part I: The Conic Tverberg’s Theorem

Abstract

Tverberg's 1966 theorem asserts that every set X of (m − 1)We give a short and elementary proof of a theorem on convex cones which generalizes this result. As a consequence, we deduce several divisibility properties, including the characterization of extremal sets which have no partition such that m i=1 conv X i is at least one-dimensional and, in the particular cases m = 3 and m = 4, the proof of Reay's conjecture that every set of (m − 1)(d + 1) + k + 1 points in general position in R d has a partition such that m i=1 conv X i is at least k-dimensional.

Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5

“…In fact, our results go further, from two different points of view. As for d = 2 [4], d = 3 [12] and m = 3 or 4 [13], we show that, in each case, the partition X 1 , X 2 , . .…”

“…In each case, we proceed by finite induction, based on divisibility lemmas derived from the conic Tverberg's theorem (see [13,Theorem 2.1]). Oriented matroids [5] are also intensively used, and appear as a suitable and natural tool (more adapted, we believe, than vertex figures or Gale diagrams, which could constitute an alternative for certain lemmas).…”

“…, X m such that m i=1 conv X i is at least k-dimensional. Using the tools developed in [13] and oriented matroid theory, we prove this conjecture for d = 4 and d = 5. How about, to that end, we introduce the notion of a k-lopsided oriented matroid and we characterize these combinatorial objects for certain values of k. Divisibility properties for subsets of R d with other independence conditions are also obtained, thus settling several particular cases of a generalization of Reay's conjecture.…”

“…Following the usual terminology [10], a subset X of R d is said to be (m, In the first part of this paper [13], we have given new proofs of these two theorems and characterized the sets of R d with 2d(m − 1) + 1 elements which are not (m, 1)-divisible. For k ≥ 2, a cardinality condition on |X | is no longer sufficient for (m, k)-divisibility.…”

Helge Tverberg is eighty: A personal tribute

“…This innocent looking statement is a generalization of Radon's theorem which deals with the case r = 2 and for which there is an easy proof. Several proofs of Tverberg's theorem are known but even the simplest ones by Sarkaria [7] and Roudneff [6] are not easy. Tverberg himself gave a second proof [11].…”