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Polyhedral-type approximations of convex-like domains in $\mathbb{C}^d$ havebeen considered recently by the second author. In particular, the decay rate ofthe error in optimal volume approximation as a function of the number of facetshas been obtained. In this article, we take these studies further byinvestigating polyhedra constructed using random points (Poisson or binomialprocess) on the boundary of a strongly $\mathbb{C}$-convex domain. We determinethe rate of error in volume approximation of the domain by random polyhedra,and conjecture the precise value of the minimal limiting constant. Analogous tothe real case, the exponent appearing in the error rate of random volumeapproximation coincides with that of optimal volume approximation, and can beinterpreted in terms of the Hausdorff dimension of a naturally-occurring metricspace. Moreover, the limiting constant is conjectured to depend on theM\"obius-Fefferman measure, which is a complex analogue of the Blaschke surfacearea measure. Finally, we also prove $L^1$-convergence, variance bounds, andnormal approximation.

Volume Approximation of Strongly ${\mathbb C}$-Convex Domains by Random Polyhedra