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

TL;DR

This paper studies how well random polyhedra approximate strongly -convex domains in complex space, providing error rates, conjectures on constants, and proving convergence and distributional results.

Contribution

It extends previous work by analyzing volume approximation errors using random boundary points and conjectures the limiting constants in complex convex geometry.

Findings

Error rate of volume approximation matches optimal case exponent.

Conjectured limiting constant depends on Mb4obius-Fefferman measure.

Proved convergence, variance bounds, and normal approximation for the approximation process.

Abstract

Polyhedral-type approximations of convex-like domains in $\mathbb{C}^d$ 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 $\mathbb{C}$-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 M\"obius-Fefferman measure, which is a complex analogue of the Blaschke surface area measure. Finally, we also prove $L^1$-convergence, variance bounds, and normal approximation.