Statistical mechanics of convex bodies

The difficulties inherent in the construction of two-dimensional pressure ensembles are discussed, and are tackled by defining an energy cost depending on the convex hull of the set of particles. An energy proportional to the area of the convex hull is not able to prevent evaporation of the system, whereas an energy proportional to the area of the circumcircle of the convex hull ensures a thermodynamic behavior. In the latter model, which turns out to be exactly solvable, various characterizations are given of the geometry of a typical state.