Limit shape of convex lattice polygons having the minimal L∞ diameter wrt the number of their vertices

This paper deals with the class of optimal convex lattice polygons having the minimal L-infinity-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) = (pi/root 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-infinity-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-infinity-diameter, equal to m(n), then the sequence of normalized polygons (1/diam(infinity)(P-n)).P-n=(1/m(n)).P-n tends to the curve y(2) = (1/2 - root 1 - 2\x\ - \x\)(2), where x is an element of [-1/2, 1/2], as n-->infinity. (C) 1998 Elsevier Science B.V. All rights reserved.