# Proportional concentration phenomena on the sphere

@article{Artstein2002ProportionalCP, title={Proportional concentration phenomena on the sphere}, author={S. Artstein}, journal={Israel Journal of Mathematics}, year={2002}, volume={132}, pages={337-358} }

In this paper we establish concentration phenomena for subspaces with arbitrary dimension. Namely, we display conditions under which the Haar measure on the sphere concentrates on a neighborhood of the intersection of the sphere with a subspace ofRn of a given dimension. We display applications to a problem of projections of points on the sphere, and to the duality of entropy numbers conjecture.

#### 29 Citations

More on the duality conjecture for entropy numbers

- Mathematics
- 2003

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci.… Expand

Exponential and Gaussian concentration of 1-Lipschitz maps

- Mathematics
- 2009

In this paper, we prove an exponential and Ganssian concentration inequality for 1-Lipschitz maps from mm-spaces to Hadamard manifolds. In particular, we give a complete answer to a question by M.… Expand

Asymptotic formulas for the diameter of sections of symmetric convex bodies

- Mathematics
- 2005

Sharpening work of the first two authors, for every proportion λ∈(0,1) we provide exact quantitative relations between global parameters of n-dimensional symmetric convex bodies and the diameter of… Expand

Duality of metric entropy in Euclidean space

- Mathematics
- 2003

Abstract Let K be a convex body in a Euclidean space, K ° its polar body and D the Euclidean unit ball. We prove that the covering numbers N ( K , tD ) and N ( D , tK °) are comparable in the… Expand

A Bernstein—Chernoff deviation inequality, and geometric properties of random families of operators

- Mathematics
- 2006

In this paper we first describe a new deviation inequality for sums of independent random variables which uses the precise constants appearing in the tails of their distributions, and can reflect in… Expand

Geometric Applications of Chernoff-Type Estimates

- Mathematics
- 2007

In this paper we present a probabilistic approach to some geometric problems in asymptotic convex geometry. The aim of this paper is to demonstrate that the well known Chernoff bounds from… Expand

On the isotropic constant of random polytopes

- Mathematics
- 2010

Let K be an isotropic 1-unconditional convex body in R. For every N > n consider N independent random points x1, . . . , xN uniformly distributed in K. We prove that, with probability greater than 1… Expand

From Steiner Formulas for Cones to Concentration of Intrinsic Volumes

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2014

A systematic technique for studying conic intrinsic volumes using methods from probability, based on a general Steiner formula for cones, which leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality. Expand

Non-Asymptotic Theory of Random Matrices Lecture 13 : Sections of convex sets via entropy and volume

- 2007

Recall the Low M-estimate from the previous lecture: Let T ⊆ R n be convex and symmetric, and let E be a random subspace of R n , with codimension k. Then, diam(T ∩ E) ≤ C l (T) √ k , with high… Expand

Ju l 2 00 4 Duality of Metric Entropy ∗

- 2004

For two convex bodies K and T in R n , the covering number of K by T , denoted N (K, T), is defined as the minimal number of translates of T needed to cover K. Let us denote by K • the polar body of… Expand

#### References

SHOWING 1-10 OF 15 REFERENCES

Entropy and Asymptotic Geometry of Non-Symmetric Convex Bodies☆

- Mathematics
- 2000

Abstract We extend to the general, not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very… Expand

Subspaces of small codimension of finite-dimensional Banach spaces

- Mathematics
- 1986

Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking… Expand

Chapter 17 - Euclidean Structure in Finite Dimensional Normed Spaces

- Mathematics
- 2001

This chapter discusses the results that stand between geometry, convex geometry, and functional analysis. The chapter describes the family of n -dimensional normed spaces and discusses the study on… Expand

Asymptotic Theory Of Finite Dimensional Normed Spaces

- Mathematics
- 1986

The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.-… Expand

Large Deviations and Applications

- Mathematics
- 1984

Large Deviations Cramer's Theorem Multidimensional Version of Cramer's Theorem An Infinite Dimensional Example: Brownian Motion The Ventcel-Freidlin Theory The Exit Problem Empirical Distributions… Expand

A new approach to several results of V. Milman.

- Mathematics
- 1989

ck(t7) = inf{||t>|S|| | Sc:E codimS<k}. Similarly for any subspace S a F, let us denote by qs : F — > F/S the quotient mapping. Then we denote O\\ \Sc:FdimS<k}. In this paper, we denote by l\ the n… Expand

A dozen de Finetti-style results in search of a theory

- Mathematics
- 1987

Les k premieres coordonnees d'un point uniformement distribue sur la sphere de dimension n se comportent comme des variables gaussiennes reduites independantes quand n→∞ avec k fixe. Si k→∞ le… Expand

A normal approximation for beta and gamma tail probabilities

- Mathematics
- 1984

SummaryNormal approximations are developed for the beta- and related distributions, using an approach similar to that of Peizer and Pratt (1968). No series expansions are involved, and the few… Expand