Limit shape of convex lattice polygons with minimal perimeter

Let n be a positive integer and ∥·∥ any norm in R2. Denote by B the unit ball of ∥·∥ and PB,n the class of convex lattice polygons with n vertices and least ∥·∥-perimeter. We prove that after suitable normalization, all members of PB,n tend to a fixed convex body, as n→∞.