Limit shape of convex lattice polygons having the minimal L∞ diameter w.r.t. the number of their vertices

Abstract

This paper deals with the class of optimal convex lattice polygons having the minimal L∞-diameter with respect to the number of their vertices. It is an already known result, that if P is a convex lattice polygon, with n vertices, then the minimal size of a squared integer grid in which P can be inscribed, is m(n) = (π/√432) n3/2 + script O sign(n log n). The known construction of the optimal polygons is implicit. The optimal convex lattice n-gon is determined uniquely only for certain values of n, but in general, there can be many different optimal polygons with the same number of vertices and the same L∞-diameter. The purpose of this paper is to show the existence and to describe the limit shape of this class of optimal polygons. It is shown that if Pn is an arbitrary sequence of optimal convex lattice polygons, having the minimal possible L∞-diameter, equal to m(n), then the sequence of normalized polygons (1/diam∞(Pn)) · Pn = (1/m(n)) · Pn tends to the curve y2 = (1/2 - √1 - 2\x\ - |x|)2, where x ∈ [-1/2, 1/2], as n → ∞.