Tverberg theorems over discrete sets of points

2018/03/05 by De Loera, Jesús A., Hogan, Thomas A., Meunier, Frédéric +1
#52 #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1803.01816

Abstract

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 We determine the m-Tverberg number, when m ≥ 3, of any discrete subset S of ℝ² (a generalization of an unpublished result of J.-P. Doignon).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.

Related

• Quantitative Tverberg theorems over lattices and other discrete sets
• A generalization of Tverberg's Theorem
• Robust Tverberg and colorful Carathéodory results via random choice
• Tverberg's theorem is 50 years old: a survey
• Algorithms for Tverberg's theorem via centerpoint theorems