Volume Approximations of Strictly Pseudoconvex Domains

In convex geometry, the Blaschke surface area measure on the boundary of a convex domain can be interpreted in terms of the complexity of approximating polyhedra. In response to a question raised by D. Barrett, this approach is formulated in the holomorphic setting to establish an alternate interpretation of Fefferman's hypersurface measure on boundaries of strictly pseudoconvex domains in $\mathbb{C}^2$. In particular, it is shown that Fefferman's measure can be recovered from the Bergman kernel of the domain. A connection with the geometry of the Heisenberg group, emerging from these results, is also discussed.