This preprint was accepted January, 1998.
We provide a full large deviation principle (LDP) for the uniform measure on certain ensembles of convex lattice polygons. This LDP provides for the analysis of concentration of the measure on convex closed curves. In particular, convergence to a limiting shape results in some particular cases, including convergence to a circle when the ensemble is defined as those centered convex polygons with total length bounded by a constant. The Gauss-Minkowskii transform of convex curves plays a crucial role in our approach.