Volume approximation of strongly C-convex domains by random polyhedra

Athreya, S and Gupta, P and Yogeshwaran, D (2023) Volume approximation of strongly C-convex domains by random polyhedra. In: Advances in Mathematics, 432 .

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Abstract

Polyhedral-type approximations of convex-like domains in Cd 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. © 2023 Elsevier Inc.

Item Type:	Journal Article
Publication:	Advances in Mathematics
Publisher:	Academic Press Inc.
Additional Information:	The copyright for this article belongs to the Academic Press Inc.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	10 Nov 2023 10:15
Last Modified:	10 Nov 2023 10:15
URI:	https://eprints.iisc.ac.in/id/eprint/83189

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