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

2022/03/23 by Siva Athreya, Athreya, Siva, Purvi Gupta +3
Mathematics · #32F17 #60D05 #60F05 #60G55 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Approximation and Integration #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2203.12444

openalex publication_date 2022/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Polyhedral-type approximations of convex-like domains in ℂ^d have been considered recently by the second author. In particular, the decay rate of the error in optimal volume approximation as a function of the number of facets has been obtained. In this article, we take these studies further by investigating polyhedra constructed using random points (Poisson or binomial process) on the boundary of a strongly ℂ-convex domain. We determine the rate of error in volume approximation of the domain by random polyhedra, and conjecture the precise value of the minimal limiting constant. Analogous to the real case, the exponent appearing in the error rate of random volume approximation coincides with that of optimal volume approximation, and can be interpreted in terms of the Hausdorff dimension of a naturally-occurring metric space. Moreover, the limiting constant is conjectured to depend on the Möbius-Fefferman measure, which is a complex analogue of the Blaschke surface area measure. Finally, we also prove L¹-convergence, variance bounds, and normal approximation.

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