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Notes on Optimal Convex Lattice Polygons
Author(s) -
Žunić Joviša
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398004482
Subject(s) - mathematics , combinatorics , regular polygon , polygon (computer graphics) , convex polygon , star shaped polygon , perimeter , lattice (music) , upper and lower bounds , convex set , geometry , convex optimization , mathematical analysis , telecommunications , physics , frame (networking) , computer science , acoustics
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] 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] where s is the perimeter of such a polygon. This result implies that the area of these polygons is asymptotically less than 0.0191612. 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 is15 784 ⋅ n 3 ≈ 0.0191326 ⋅ n 3 . The previous upper bound for the minimal area of a convex lattice n‐gon is improved to1 54 ⋅ n 3 ≈ 0.0185185 ⋅ n 3as n →∞. 1991 Mathematics Subject Classification 11H06, 52C05, 05D05.