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

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)=2π54Apn3/2+O(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 thatNp(s)=3Ap32π23s2/3+O(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 Jarnikʹs 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 O(n log n) and O(s1/3 log s), respectively.