A probabilistic approach to the geometry of the ℓpn-ball

A probabilistic approach to the geometry of the ℓ_pⁿ-ball

Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor

Source: Ann. Probab. Volume 33, Number 2 (2005), 480-513.

Abstract

This article investigates, by probabilistic methods, various geometric questions on B_pⁿ, the unit ball of ℓ_pⁿ. We propose realizations in terms of independent random variables of several distributions on B_pⁿ, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B_pⁿ. As another application, we compute moments of linear functionals on B_pⁿ, which gives sharp constants in Khinchine’s inequalities on B_pⁿ and determines the ψ₂-constant of all directions on B_pⁿ. We also study the extremal values of several Gaussian averages on sections of B_pⁿ (including mean width and ℓ-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in ℓ₂ and to covering numbers of polyhedra complete the exposition.

First Page: Show

Primary Subjects: 60E15, 52A20, 52A38, 52A40

Keywords: ℓ_p^n-ball; Gaussian measure; extremal sections

Full-text: Open access

Screen Optimized PDF File (249 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1109868589
Digital Object Identifier: doi:10.1214/009117904000000874
Mathematical Reviews number (MathSciNet): MR2123199

back to Table of Contents

References

Antilla, M., Ball, K. and Perissinaki, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 4723--4735.

Mathematical Reviews (MathSciNet): MR1997580

Digital Object Identifier: doi:10.1090/S0002-9947-03-03085-X

Zentralblatt MATH: 1033.52003

Ball, K. (1986). Cube slicing in $\mathbbR^n$. Proc. Amer. Math. Soc. 97 465--473.

Mathematical Reviews (MathSciNet): MR840631

Ball, K. (1989). Volumes of sections of cubes and related problems. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1376 251--260. Springer, New York.

Mathematical Reviews (MathSciNet): MR1008726

Zentralblatt MATH: 0674.46008

Ball, K. and Perissinaki, I. (1998). The subindependence of coordinate slabs in $\ell_p^n$ balls. Israel J. Math. 107 289--299.

Mathematical Reviews (MathSciNet): MR1658571

Banaszczyk, W. (1998). Balancing vectors and Gaussian measures on $n$-dimensional convex bodies. Random Structures Algorithms 12 351--360.

Mathematical Reviews (MathSciNet): MR1639752

Barthe, F. (2001). Extremal properties of central half-spaces for product measures. J. Funct. Anal. 182 81--107.

Mathematical Reviews (MathSciNet): MR1829243

Digital Object Identifier: doi:10.1006/jfan.2000.3708

Zentralblatt MATH: 0984.28003

Barthe, F., Csörnyei, M. and Naor, A. (2003). A note on simultaneous polar and Cartesian decomposition. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 1--19. Springer, New York.

Mathematical Reviews (MathSciNet): MR2083383

Zentralblatt MATH: 1036.52004

Barthe, F. and Koldobsky, A. (2003). Extremal slabs in the cube and the Laplace transform. Adv. Math. 174 89--114.

Mathematical Reviews (MathSciNet): MR1959893

Digital Object Identifier: doi:10.1016/S0001-8708(02)00055-5

Zentralblatt MATH: 1037.52006

Barthe, F. and Naor, A. (2002). Hyperplane projections of the unit ball of $\ell_p^n$. Discrete Comput. Geom. 27 215--226.

Mathematical Reviews (MathSciNet): MR1880938

Bobkov, S. G. and Nazarov, F. L. (2003). Large deviations of typical linear functionals on a convex body with unconditional basis. Preprint.

Mathematical Reviews (MathSciNet): MR2073422

Bobkov, S. G. and Nazarov, F. L. (2003). On convex bodies and log-concave probability measures with unconditional basis. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 53--69. Springer, New York.

Mathematical Reviews (MathSciNet): MR2083388

Zentralblatt MATH: 1039.52004

Bourgain, J. (1991). On the distribution of polynomials on high dimensional convex sets. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1469 127--137. Springer, New York.

Mathematical Reviews (MathSciNet): MR1122617

Zentralblatt MATH: 0773.46013

Brascamp, H. J. and Lieb, E. H. (1976). Best constants in Young's inequality, its converse and its generalization to more than three functions. Adv. Math. 20 151--173.

Mathematical Reviews (MathSciNet): MR412366

Digital Object Identifier: doi:10.1016/0001-8708(76)90184-5

Zentralblatt MATH: 0339.26020

Carl, B. and Pajor, A. (1988). Gelfand numbers of operators with values in a Hilbert space. Invent. Math. 94 479--504.

Mathematical Reviews (MathSciNet): MR969241

Digital Object Identifier: doi:10.1007/BF01394273

Fang, K. T. and Zhang, Y. T. (1990). Generalized Multivariate Analysis. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1079542

Zentralblatt MATH: 0724.62054

Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89--103.

Mathematical Reviews (MathSciNet): MR309498

Digital Object Identifier: doi:10.1007/BF01651330

Zentralblatt MATH: 0346.06011

Gluskin, E. D. and Kwapień, S. (1995). Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114 303--309.

Mathematical Reviews (MathSciNet): MR1338834

Kannan, R., Lovász, L. and Simonovits, M. (1997). Random walks and an $O^*(n^5)$ volume algorithm for convex bodies. Random Structures Algorithms 11 1--50.

Mathematical Reviews (MathSciNet): MR1608200

Kanter, M. (1977). Unimodality and dominance for symmetric random vectors. Trans. Amer. Math. Soc. 229 65--85.

Mathematical Reviews (MathSciNet): MR445580

Koldobsky, A. (1998). An application of the Fourier transform to sections of star bodies. Israel J. Math. 106 157--164.

Mathematical Reviews (MathSciNet): MR1656857

Ledoux, M. and Talagrand M. (1991). Probability in Banach Spaces. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1102015

Zentralblatt MATH: 0748.60004

Meyer, M. and Pajor, A. (1988). Sections of the unit ball of $\ell^n_p$. J. Funct. Anal. 80 109--123.

Mathematical Reviews (MathSciNet): MR960226

Digital Object Identifier: doi:10.1016/0022-1236(88)90068-7

Zentralblatt MATH: 0667.46004

Milman, V. and Pajor, A. (1989). Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1376 64--104. Springer, New York.

Mathematical Reviews (MathSciNet): MR1008717

Zentralblatt MATH: 0679.46012

Naor, A. (2004). The surface measure and cone measure on the sphere of $\ell_p^n$. Trans. Amer. Math. Soc. To appear.

Mathematical Reviews (MathSciNet): MR2262841

Zentralblatt MATH: 1109.60006

Digital Object Identifier: doi:10.1090/S0002-9947-06-03939-0

Naor, A. and Romik, D. (2003). Projecting the surface measure of the sphere of $\ell_p^n$. Ann. Inst. H. Poincaré Probab. Statist. 39 241--261.

Mathematical Reviews (MathSciNet): MR1962135

Digital Object Identifier: doi:10.1016/S0246-0203(02)00008-0

Pajor, A. and Tomczak-Jaegermann, N. (1986). Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Amer. Math. Soc. 97 637--642.

Mathematical Reviews (MathSciNet): MR845980

Paouris, G. (2003). $\Psi_2$-estimates for linear functionals on zonoids. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 211--222. Springer, New York.

Mathematical Reviews (MathSciNet): MR2083399

Zentralblatt MATH: 1036.52009

Pietsch, A. (1978). Operator Ideals. VEB, Berlin.

Mathematical Reviews (MathSciNet): MR519680

Pisier, G. (1989). The Volume of Convex Bodies and Banach Space Geometry. Cambridge Univ. Press.

Mathematical Reviews (MathSciNet): MR1036275

Zentralblatt MATH: 0698.46008

Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Sci. Math. 34 335--343.

Mathematical Reviews (MathSciNet): MR404557

Rachev, S. T. and Rüschendorf, L. (1991). Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19 1311--1337.

Mathematical Reviews (MathSciNet): MR1112418

JSTOR: links.jstor.org

Schechtman, G. and Zinn, J. (1990). On the volume of intersection of two $L_p^n$ balls. Proc. Amer. Math. Soc. 110 217--224.

Mathematical Reviews (MathSciNet): MR1015684

Schechtman, G. and Zinn, J. (2000). Concentration on the $\ell_p^n$ ball. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 245--256. Springer, New York.

Mathematical Reviews (MathSciNet): MR1796723

Zentralblatt MATH: 0971.46009

Schütt, C. (1983). Entropy numbers of diagonal operators between symmetric spaces. J. Approx. Theory 40 121--128.

Mathematical Reviews (MathSciNet): MR732693

Digital Object Identifier: doi:10.1016/0021-9045(84)90021-2

Zentralblatt MATH: 0497.41017

Sudakov, V. N. (1971). Gaussian random processes and measures of solid angles in Hilbert space. Soviet Math. Dokl. 12 412--415.

Mathematical Reviews (MathSciNet): MR288832

Widder, D. V. (1941). The Laplace Transform. Princeton Univ. Press.

Mathematical Reviews (MathSciNet): MR5923

previous :: next