New cases of Reay’s conjecture on partitions of points into simplices with -dimensional intersection

Reay’s conjecture asserts that every set of ( m − 1 ) ( d + 1 ) + k + 1 points in general position in R d (with 0 ≤ k ≤ d ) has a partition X 1 , X 2 , … , X m such that ⋂ i = 1 m  conv X i is at least k -dimensional. We prove this conjecture in several cases: when m ≤ 8 (for arbitrary d and k ), when d = 6 , d = 7 and d = 8 (for arbitrary m and k ), and when k = 1 and d ≤ 24 (for arbitrary m ).