Tight bounds on discrete quantitative Helly numbers

TL;DR

This paper establishes tight bounds on the discrete Helly numbers for various sets, especially integer lattices, by providing combinatorial characterizations and exact values for small cases, advancing understanding in discrete convex geometry.

Contribution

It introduces a combinatorial description of c(S,k), improves bounds for general sets, and determines exact values for small k in Z^n, closing previous gaps in the theory.

Findings

c(Z^n,k) = Theta(k^{(n-1)/(n+1)}) for fixed n

Improved bounds for c(S,k) for general discrete sets

Exact values of c(Z^n,k) for k <= 4

Abstract

Given a subset S of R^n, let c(S,k) be the smallest number t such that whenever finitely many convex sets have exactly k common points in S, there exist at most t of these sets that already have exactly k common points in S. For S = Z^n, this number was introduced by Aliev et al. [2014] who gave an explicit bound showing that c(Z^n,k) = O(k) holds for every fixed n. Recently, Chestnut et al. [2015] improved this to c(Z^n,k) = O(k (log log k)(log k)^{-1/3} ) and provided the lower bound c(Z^n,k) = Omega(k^{(n-1)/(n+1)}). We provide a combinatorial description of c(S,k) in terms of polytopes with vertices in S and use it to improve the previously known bounds as follows: We strengthen the bound of Aliev et al. [2014] by a constant factor and extend it to general discrete sets S. We close the gap for Z^n by showing that c(Z^n,k) = Theta(k^{(n-1)/(n+1)}) holds for every fixed n. Finally, we determine the exact values of c(Z^n,k) for all k <= 4.