2002
DOI: 10.1090/s0002-9947-02-02962-8
|View full text |Cite
|
Sign up to set email alerts
|

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.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1
1

Citation Types

2
68
0

Year Published

2005
2005
2023
2023

Publication Types

Select...
5
2
1

Relationship

1
7

Authors

Journals

citations
Cited by 52 publications
(70 citation statements)
references
References 33 publications
2
68
0
Order By: Relevance
“…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.…”
Section: Random Approximation Of Polytopes By Polytopesmentioning
confidence: 65%
See 1 more Smart Citation
“…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.…”
Section: Random Approximation Of Polytopes By Polytopesmentioning
confidence: 65%
“…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.…”
Section: Random Approximation Of Convex Bodies By Polytopesmentioning
confidence: 84%
“…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.…”
Section: The Main Theoremmentioning
confidence: 76%
“…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].…”
Section: Santal ó S-regionsmentioning
confidence: 99%