Notes on optimal convex lattice polygons

Abstract

In this paper we consider connections between three classes of optimal (in different senses) convex lattice polygons. A classical result is that if G is a strictly convex curve of length s, then the maximal number of integer points lying on G is essentially 3/(2π)1/3 · s2/3 ≈ 1.625578 · s2/3. It is proved here that members of a class of digital convex polygons which have the maximal number of vertices with respect to their diameter are good approximations of these curves. We show that the number of vertices of these polygons is asymptotically 12/(2π(2+√2·In(1+√2)))2/3· s2/31.60739·s2/3, where s is the perimeter of such a polygon. This result implies that the area of these polygons is asymptotically less than 0.0191612·n3, where n is the number of vertices of the observed polygon. This result is very close to the result given by Colbourn and Simpson, which is 13/784·n3 ≈ 0.0191326·n3. The previous upper bound for the minimal area of a convex lattice n-gon is improved to 1/54·n3 ≈ 0.0185185·n3 as n → ∞.