Limit shape of optimal convex lattice polygons in the sense of different metrics

Abstract

The limit shape of optimal convex lattice polygons in the sense of different metrics is studied. A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are less than π radians. Limit shapes of the three arcs of optimal convex lattice polygons are the same curves rotated for π/2, π and 3π/2 radians and translated to form a closed curve.