Abstract: Several results in convex geometry establish asymptotic estimates for the gap between a convex domain and approximating polyhedra of increasing complexity. Is it possible to do the same for approximations of pseudoconvex domains by analytic polyhedra? In this talk, I will discuss a class of polyhedral objects in strongly pseudoconvex domains that allow for such estimates. Connections with both the Fefferman hypersurface measure and a tiling problem on the Heisenberg group will be discussed. I will also indicate how our formulation suggests a way to discuss volume approximations of more general pseudoconvex domains.