Extremal problems on convex lattice polygons in sense of lp-metrics

Abstract

This paper expresses the minimal possible lp-perimeter of a convex lattice polygon with respect to its number of vertices, where p is an arbitrary integer or p = ∞. It will be shown that such a number, denoted by sp(n), has n3/2 as the order of magnitude for any choice of p. Moreover, sp(n) = 2π/√54Apn3/2 + (n), where n is the number of vertices, Ap equals the area of planar shape \x\p + \y\p < 1, and p is an integer greater than 1. A consequence of the previous result is the solution of the inverse problem. It is shown that Np(s)=33√Ap/3√2π2s2/3 + (s1/3) equals the maximal possible number of vertices of a convex lattice polygon whose lp-perimeter is equal to s. The latter result in a particular case p=2 follows from a well known Jarnik's result. The method used cannot be applied directly to the cases p = 1 and ∞. A slight modification is necessary. In the obtained results the leading terms are in accordance with the above formulas (A1 =2 and A∞ =4), while the rest terms in the expressions for sp(n) and Np(s) are replaced with (n log n) and (s1/3 logs), respectively.