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

Tverberg’s 1966 theorem asserts that every set X of (m − 1)(d + 1) + 1 points in Rdhas a partition X1, X2, . . . , Xmsuch that∩i = 1mconvXi ≠ = φ. We give a short and elementary proof of a theorem on convex cones which generalizes this result. As a consequence, we deduce several divisibility properties, including the characterization of extremal sets which have no partition such that∩i = 1mconvXiis at least one-dimensional and, in the particular cases m = 3 and m = 4, the proof of Reay’s conjecture that every set of (m − 1)(d + 1) + k + 1 points in general position in Rdhas a partition such that∩i = 1mconvXiis at least k -dimensional.