A fractional Helly theorem for convex lattice sets
Let \({\mathbb Z}^d\) denote the integer lattice in Euclidean \(d\)-space \({\mathbb R}^d\). The intersection of any convex set of \({\mathbb R}^d\) with \({\mathbb Z}^d\) is called a convex lattice set. The authors prove that if for every \(\alpha \in (0,1]\) there is a positive number \(\beta\) such that for arbitrary convex lattice sets \(F_1, \dots ,F_n\) in \({\mathbb Z} ^d\) with \(\bigcap_{i\in I} F_i \not = \emptyset\) for at least \(\alpha {n \choose d+1}\) index sets \(I \subset \{1, \dots,n\}\) of size \(d+1\), then at least \(\beta n\) of the sets \(F_1,\dots,F_n\) have a common point.
scientific article; zbMATH DE number 1903515