Please use this identifier to cite or link to this item: https://open.uns.ac.rs/handle/123456789/12744
Title: Notes on optimal convex lattice polygons
Authors: Žunić J.
Issue Date: 1-Jan-1998
Journal: Bulletin of the London Mathematical Society
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 → ∞.
URI: https://open.uns.ac.rs/handle/123456789/12744
ISSN: 246093
DOI: 10.1112/S0024609398004482
Appears in Collections:PMF Publikacije/Publications

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