NOTES ON OPTIMAL CONVEX LATTICE POLYGONS

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 num ber of integer points lying on G is essentially 3/(2 pi)(1/3).s(2/3) a pproximate to 1.62578.s(2/3). It is proved here that members of a clas s of digital convex polygons which have the maximal number of vertices with respect to their diameter are good approximations of these curve s. We show that the number of vertices of these polygons is asymptotic ally 12/(2 pi(2+root 2.1n(1+root 2)))(2/3).s(2/3) approximate to 1.607 39.s(2/3), where s is the perimeter of such a polygon. This result imp lies that the area of these polygons is asymptotically less than 0.019 1612.n(3), 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 15/784.n(3) approximate to 0.0191326.n(3). The previous uppe r bound for the minimal area of a convex lattice n-gon is improved to 1/54. n(3) approximate to 0.0185185.n(3) as n --> infinity.