On Finding The Limit Shape Of Optimal Convex Lattice Polygons

Classes of convex lattice polygons which have minimal lp-perimeter with respect to the number of their vertices are said to be optimal in sense of lp metric. rpose 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. proved that if p is arbitrary integer or oo, the limit shape of the southeast arc of optimal convex lattice polygons in sense of lp 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. shapes of the other three arcs of optimal convex lattice polygons are the same curves (γp) rotated for π/2, π and 3π/2 radians and translated to form a closed curve.