The perimeter of optimal convex lattice polygons in the sense of different metrics

Classes of convex lattice polygons which have minimal l(p)-perimeter with respect to the number of their vertices are said to be optimal in the sense of the l(p)-metric. Tt is proved that if p and q are arbitrary integers or infinity, the asymptotic expression for the l(p)-perimeter of these optimal convex lattice polygons Q(p)(n) as a function of the number of their vertices n is per(q)(Q(p)(n)) = C(p)(q)pi/root 6A(p)(3) n(3/2) + O(n(1+epsilon)) for arbritary epsilon > 0, where C-p(q) = integral integral (/x/p+/y/p less than or equal to1) (q)root /x/(q) + /y/(q) dx dy, and A(p) is equal to the area of the planar shape \x\(p) + \y\(p) less than or equal to 1.