Abstract
Given a set S subset of R-2, 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 R(2 )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 {alpha(n): n is an element of N-0}(2) are finite for every alpha > 1 and we deter-mine their exact values in some instances. In particular, we obtain H({2(n): n is an element of N-0}(2)) = 5, solving a problem posed by Dillon (2021).For real numbers alpha, beta > 1, we also fully characterize exponential lattices L(alpha, beta) = {alpha(n) : n is an element of N-0} x {beta(n) : n is an element of N-0} with finite Helly numbers by showing that H(L(alpha, beta)) is finite if and only if log(alpha)(beta) is rational.(c) 2023 Elsevier Ltd.
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