Properties of a curve whose convex hull covers a given convex body

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Advances in Applied Probability ◽

10.1017/s0001867800050369 ◽

2010 ◽

Vol 42 (03) ◽

pp. 605-619

Author(s):

I. Bárány ◽

F. Fodor ◽

V. Vígh

Keyword(s):

Convex Body ◽

Convex Hull ◽

Convex Bodies ◽

Upper Bounds ◽

Smooth Convex ◽

Lower And Upper Bounds ◽

Covering Theorem ◽

Intrinsic Volumes ◽

Large Numbers ◽

Distribution Matching

LetKbe ad-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote byKnthe convex hull ofnpoints chosen randomly and independently fromKaccording to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of thesth intrinsic volumesVs(Kn) ofKnfors∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes ofKn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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On denseness of certain norms in Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017640 ◽

1996 ◽

Vol 54 (2) ◽

pp. 183-196 ◽

Cited By ~ 1

Author(s):

M. Jiménez Sevilla ◽

J.P. Moreno

Keyword(s):

Convex Body ◽

Convex Hull ◽

Banach Spaces ◽

Convex Bodies ◽

Intersection Property ◽

Closed Convex Hull ◽

Locally Linear

We give several results dealing with denseness of certain classes of norms with many vertex points. We prove that, in Banach spaces with the Mazur or the weak* Mazur intersection property, every ball (convex body) can be uniformly approximated by balls (convex bodies) being the closed convex hull of their strongly vertex points. We also prove that given a countable set F, every norm can be uniformly approximated by norms which are locally linear at each point of F.

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