Abstrakt
Given a set S SUBSET OF OR EQUAL TO ℝ2, define the Helly number of S, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family ℱ of convex sets in ℝ2 such that the intersection of any N or fewer members of ℱ contains at least one point of S, there is a point of S common to all members of ℱ. We prove that the Helly numbers of exponential lattices {αn : n ELEMENT OF ℕo}2 are finite for every α > 1 and we determine their exact values in some instances.
In particular, we obtain H({2n : n ELEMENT OF ℕo}2) = 5, solving a problem posed by Dillon (2021). For real numbers α, β > 1, we also fully characterize exponential lattices L(α,β) = {αn : n ELEMENT OF ℕo} x {βn : n ELEMENT OF ℕo} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if log_α(β) is rational.