Asymptotic Hilbert polynomials and limiting shapes

The main aim of this paper is to provide a method which allows finding limiting shapes of symbolic generic initial systems of higher-dimensional subvarieties of PnPn. M. Mustaţă and S. Mayes established a connection between volumes of complements of limiting shapes and the asymptotic multiplicity for ideals of points. In the paper we prove a generalization of this fact to higher-dimensional sets.