Random points on the boundary of smooth convex bodies

Abstract

Abstract. The convex hull of n independent random points chosen on the boundary of a convex body K ⊂ R d according to a given density function is a random polytope. The expectation of its i-th intrinsic volume for i = 1, . . . , d is investigated. In the case that the boundary of K is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as n → ∞ are derived.

Best and random approximation of a convex body by a polytope

“…This clearly leads to a random polytope with N vertices. And by results of Schütt and Werner [62] and also Reitzner [50], the expected volume difference is of order N − 2 n−1 , which is smaller than the order in (7.20) as is to be expected, but also surprisingly much bigger than the order N − n n−1 occurring in Theorem 7.1.…”

“…Müller [48] proved this result for the Euclidean ball. Reitzner [50] proved this result for convex bodies with C 2 -boundary and everywhere positive curvature.…”

Floating bodies and approximation of convex bodies by polytopes

“…A crucial ingredient in the proof of this theorem were the surface bodies. This theorem was also proved in [28] under stronger smoothness assumptions on the boundary of K.…”

Geometric representation of classes of concave functions and duality

“…It proved to be fundamental in the solution of the affine Plateau problem by Trudinger and Wang [47,48], in the theory of valuations where the affine and centro-affine surface areas have been characterized by Ludwig and Reitzner [33] and Haberl and Parapatits [25] as unique valuations satisfying certain invariance properties. Affine surface area appears naturally in the approximation of general convex bodies by polytopes, e.g., [12,41,46]. Furthermore, there are connections to e.g., PDEs and ODEs and concentration of volume (e.g., [21,35]), information theory (e.g., [6,16,38,50]) and in a spherical and hyperbolic setting [9,10].…”