Limit theorems for random polytopes with vertices on convex surfaces

We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes...

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Bibliographic Details
Published inAdvances in applied probability Vol. 50; no. 4; pp. 1227 - 1245
Main Authors Turchi, N., Wespi, F.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.12.2018
Applied Probability Trust
Subjects
Apexes
Central limit theorem
Convexity
Hulls
Original Article
Polytopes
Probability
Theorems
Central limit theorem
surface body
variance
Primary 52A22
random polytope
stochastic geometry
intrinsic volume
60F05
Secondary 60D05
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ISSN0001-8678
1475-6064
DOI10.1017/apr.2018.58

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Summary:We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.
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ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2018.58