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

A longstanding conjecture of Reay asserts that every set X of (m − 1)(d + 1) + k + 1 points in general position in Rd has a partition X1, X2,⋯ , Xmsuch that∩i = 1mconvXiis at least k -dimensional. Using the tools developed in 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 Rd with other independence conditions are also obtained, thus settling several particular cases of a generalization of Reay’s conjecture.