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

Abstract

A longstanding conjecture of Reay asserts that every set X of (m−1)(d +1)+k+1 points in general position in R d has a partition X 1 , X 2 , . . . , 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.

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

“…This will become apparent in the next sections, and much more in [8], where extremal sets related to this problem are studied. Note that the obstruction can already be detected in the classical example of three triangles crossing at a point in R 2 (in other words, nine points in general, but not strongly general, position in the plane), see [3,5].…”

“…Several (m, k)-divisibility theorems, in the present paper as well as in [8], will be proved by induction on k. For that purpose, we now give sufficient conditions for an…”

“…Nevertheless, we have often given a reformulation of the geometric arguments used in the proofs in terms of oriented matroids, which may constitute a purely combinatorial treatment of many of our results. The knowledge of oriented matroid theory is not necessary here, but will be required in [8], where it becomes a useful and, we believe, very natural tool (see [2] for the basic notions concerning this theory).…”

“…It has been verified for d = 2 [3] and d = 3 [7] by purely geometric means. In [8], the divisibility lemmas of Section 3 and oriented matroid theory will be used to settle the conjecture in R 4 and R 5 . On the other hand, various methods have been applied for m = 2 (see [4,5]).…”

Enclosing Depth and other Depth Measures

“…The conjecture is known to be true when S is in so-called strongly general position [23], for general position in some dimensions [24,25,26] (see also [4] for more information), and without any assumption of general position for d ≤ 2 in an unpublished M. Sc thesis in Hebrew by Akiva Kadari (see [15]). While Kalai's conjecture is specifically about Tverberg depth, the sum of dimensions of depth regions can be computed for any depth measure, and thus the conjecture can be generalized to other depth measures.…”