Abstract
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and a Berry–Esseen bound for functionals of independent random variables.
Highlights
Introduction and main results1.1 BackgroundThe theory of random convex hulls has a long history, going back to Sylvester’s famous four-point problem [62]
In this article we focus on the convex hull of independent and identically distributed points taken from the boundary of a fixed convex body K
In [63] a central limit theorem is proven for the volume of the random inscribed polytope inside a sufficiently smooth convex body in Euclidean space
Summary
The theory of random convex hulls has a long history, going back to Sylvester’s famous four-point problem [62]. In this article we focus on the convex hull of independent and identically distributed points taken from the boundary of a fixed convex body K This model of a so-called random inscribed polytope in K was investigated in [13,16,51,52,56,57,60,64], mainly from an asymptotic point of view (as the number of points tends to infinity). Of particular interest are the cases of random inscribed polytopes inside convex bodies in spherical or hyperbolic geometry This continues a recent trend in stochastic geometry of generalizing known results to the non-Euclidean setting, and in particular to spherical and hyperbolic geometry, see e.g. Let us mention that the analogous result for the random model where points are distributed inside the convex body was proven for the Euclidean case in [53], and were recently generalized to the non-Euclidean setting in [9]
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