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NOTES ON OPTIMAL CONVEX LATTICE POLYGONS

Published online by Cambridge University Press:  01 July 1998

JOVIšA ŽUNIĆ
JOVIšA ŽUNIĆ
Affiliation:
University of Novi Sad, Faculty of Engineering, Trg D. Obradovića 6, 21000 Novi Sad, Yugoslavia
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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

formula here

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

formula here

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 15/784·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→∞.

Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 4 , July 1998 , pp. 377 - 385
Copyright
© The London Mathematical Society 1998

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