Limit shape of optimal convex lattice polygons in the sense of different metrics

摘要

Classes of convex lattice polygons which have minimal l_p-perimeter with respect to the number of their vertices are said to be optimal in the sense of l_p metric. The purpose of this paper is to prove the existence and explicitly find the limit shape of the sequence of these optimal convex lattice polygons as the number of their vertices tends to infinity. It is proved that if p is arbitrary integer or ∞, the limit shape of the south-east arc of optimal convex lattice polygons in sense of l_p metric is a curve given parametrically by (C_x~p(α)/I_p, C_y~p(α)/I_p), 0 < α < ∞, where C_x~p(α) = α/2 (-1/3(α~p + 1)~(-3/p) + Σ from k=0 to ∞(_k~(-3/p-1)) (α~(pk))/(pk = 1)), C_y~p(α) = α~2 (-1/3(α~p + 1)~(-3/p) + Σ from k=0 to ∞(_k~(-3/p-1)) (α~(pk))/(pk + 2)), I_p = ∫_0~1(p~(1-1p)~2)~(1/2) dl. Some applications of the limit shape in calculating asymptotic expressions for area of the optimal convex lattice polygons are presented.