On Finding The Limit Shape Of Optimal Convex Lattice Polygons

Abstract

Abstract Abstract 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 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 oo, the limit shape of the southeast arc of optimal convex lattice polygons in sense of l p metric is a curve given parametrically by C p x (s) I p ; C p y (s) I p ;0 < s < ∞ , where C p x (α)= 1 2 ∫ 0 α α P+1 p 1−n p p 2 − n 2 α 2 dn; C p x (α)= ∫ 0 p αP+1 α n 1−n p p − n 2 α dn; I p = ∫ 0 1 1 −lp P 2 dl. Limit shapes of the other three arcs of optimal convex lattice polygons are the same curves (γ p ) rotated for π/2, π and 3π/2 radians and translated to form a closed curve.