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

Abstract

This paper expresses the minimal possible l p -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 s p ( n), has n 3/2 as the order of magnitude for any choice of p. Moreover, s p(n)= 2π 54A p n 3/2+ O(n), where n is the number of vertices, A p 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 N p(s)= 3 A p 3 2π 2 3 s 2/3+ O(s 1/3) equals the maximal possible number of vertices of a convex lattice polygon whose l p -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 ( A 1=2 and A ∞=4), while the rest terms in the expressions for s p ( n) and N p ( s) are replaced with O(n log n) and O(s 1/3 log s) , respectively.