Tverberg theorems over discrete sets of points

This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset S ⊂ R^d and the intersection of convex hulls is required to have a non-empty intersection with S). We determine the m-Tverberg number, when m ≥ 3, of any discrete subset S of R² (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of Z³ and Z^j × R^k and an integer version of the well-known positive-fraction selection lemma of J. Pach.