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 ) n 3 / 2 + O ( 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 P n 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 ∞( P n)) · P n = (1/ m( n)) · P n tends to the curve y 2 = ( 1 2 − 1 − 2 | x | − | x | ) 2 , where x ∈ [ − 1 2 , 1 2 ] , , as n → ∞.