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

Reay’s conjecture asserts that every set of (m−1)(d+1)+k+1(m−1)(d+1)+k+1 points in general position in RdRd (with 0≤k≤d0≤k≤d) has a partition X1,X2,…,XmX1,X2,…,Xm such that ⋂i=1m convXi is at least kk-dimensional. We prove this conjecture in several cases: when m≤8m≤8 (for arbitrary dd and kk), when d=6,d=7d=6,d=7 and d=8d=8 (for arbitrary mm and kk), and when k=1k=1 and d≤24d≤24 (for arbitrary mm).