Tverberg theorems over discrete sets of points
Jesús A. De Loera, Thomas A. Hogan, Frédéric Meunier, Nabil H. Mustafa · Contemporary mathematics - American Mathematical Society · 2021
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 S \subset \mathbb {R}^d and the intersection of convex hulls is required to have a non-empty intersection with S S ). We determine the m m -Tverberg number, when m ≥ 3 m \geq 3 , of any discrete subset S S of R 2 \mathbb {R}^2 (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of Z 3 \mathbb {Z}^3 and Z j × R k \mathbb {Z}^j \times \mathbb {R}^k and an integer version of the well-known positive-fraction selection lemma of J. Pach.