Inequalities between successive minima and intrinsic volumes of a 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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Further inequalities for the (generalized) Wills functional

Communications in Contemporary Mathematics ◽

10.1142/s021919972050011x ◽

2020 ◽

pp. 2050011

Author(s):

David Alonso-Gutiérrez ◽

María A. Hernández Cifre ◽

Jesús Yepes Nicolás

Keyword(s):

Convex Body ◽

Convex Bodies ◽

General Setting ◽

Concave Function ◽

Upper Bounds ◽

Edge Length ◽

Symmetric Convex ◽

Concave Functions ◽

Intrinsic Volumes

The Wills functional [Formula: see text] of a convex body [Formula: see text], defined as the sum of its intrinsic volumes [Formula: see text], turns out to have many interesting applications and properties. In this paper, we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for [Formula: see text] in terms of the volume of [Formula: see text], as well as Brunn–Minkowski and Rogers–Shephard-type inequalities for this functional. We also show that the cube of edge-length 2 maximizes [Formula: see text] among all [Formula: see text]-symmetric convex bodies in John position, and we reprove the well-known McMullen’s inequality [Formula: see text] using a different approach.

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Successive Minima and Radii

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-041-2 ◽

2009 ◽

Vol 52 (3) ◽

pp. 380-387 ◽

Cited By ~ 4

Author(s):

Martin Henk ◽

María A. Hernández Cifre

Keyword(s):

Convex Body ◽

The Body ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

Successive Minima

AbstractIn this note we present inequalities relating the successive minima of an o-symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

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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.

Download Full-text