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 Rd has a partition X 1 , X 2 ,⋯ , X m such that ∩ i  = 1 m convX i is 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.