Random Points in Isotropic Unconditional Convex Bodies

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies K, T1,…,Ts. (a) Let ɛ ∈ (0,1) and let x1,…, xN be chosen from K. Is it true that if N ⩾ C(ɛ)n log n, then ‖ I−1NLK2∑i=1Nxi⊗xi ‖

[1]  E. Lieb,et al.  A general rearrangement inequality for multiple integrals , 1974 .

[2]  Zoltán Füredi,et al.  Computing the volume is difficult , 1986, STOC '86.

[3]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[4]  G. Pisier.  ASYMPTOTIC THEORY OF FINITE DIMENSIONAL NORMED SPACES (Lecture Notes in Mathematics 1200) , 1987 .

[5]  Zoltán Füredi,et al.  Approximation of the sphere by polytopes having few vertices , 1988 .

[6]  V. Milman,et al.  On a geometric inequality , 1988 .

[7]  B. Carl,et al.  Gelfand numbers of operators with values in a Hilbert space , 1988 .

[8]  V. Milman,et al.  Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space , 1989 .

[9]  E. Gluskin.  EXTREMAL PROPERTIES OF ORTHOGONAL PARALLELEPIPEDS AND THEIR APPLICATIONS TO THE GEOMETRY OF BANACH SPACES , 1989 .

[10]  Alain Pajor,et al.  Convex bodies with few faces , 1989 .

[11]  Gideon Schechtman,et al.  On the Volume of the Intersection of Two L n p Balls , 1989 .

[12]  S. Montgomery-Smith.  The distribution of Rademacher sums , 1990 .

[13]  G. Schechtman,et al.  Another remark on the volume of the intersection of two L p n balls , 1991 .

[14]  Martin E. Dyer,et al.  Volumes Spanned by Random Points in the Hypercube , 1992, Random Struct. Algorithms.

[15]  Gideon Schechtman,et al.  An 'isomorphic' version of Dvoretzky's theorem , 1995 .

[16]  Simeon Alesker,et al.  ψ2-Estimate for the Euclidean Norm on a Convex Body in Isotropic Position , 1995 .

[17]  M. Rudelson.  Random Vectors in the Isotropic Position , 1996, math/9608208.

[18]  Miklós Simonovits,et al.  Random walks and an O*(n5) volume algorithm for convex bodies , 1997, Random Struct. Algorithms.

[19]  M. Simonovits,et al.  Random walks and an O * ( n 5 ) volume algorithm for convex bodies , 1997 .

[20]  The subindependence of coordinate slabs inlpn balls , 1998 .

[21]  Gideon Schechtman,et al.  An Isomorphic Version of Dvoretzky's Theorem, II , 1998 .

[22]  M. Schmuckenschläger.  Volume of intersections and sections of the unit ball of ℓⁿ_ , 1998 .

[23]  V. Milman,et al.  Concentration Property on Probability Spaces , 2000 .

[24]  V. Milman,et al.  Randomizing properties of convex high-dimensional bodies and some geometric inequalities , 2002 .

[25]  Keith Ball,et al.  The central limit problem for convex bodies , 2003 .

[26]  Sergey G. Bobkov,et al.  On the Central Limit Property of Convex Bodies , 2003 .

[27]  Fedor Nazarov,et al.  Large Deviations of Typical Linear Functionals on a Convex Body with Unconditional Basis , 2003 .

[28]  A. Giannopoulos,et al.  Volume radius of a random polytope in a convex body , 2003, Mathematical Proceedings of the Cambridge Philosophical Society.

[29]  Fedor Nazarov,et al.  On Convex Bodies and Log-Concave Probability Measures with Unconditional Basis , 2003 .

[30]  Concentration of mass and central limit properties of isotropic convex bodies , 2004 .

[31]  Geometric Probability and Random Cotype 2 , 2004 .

[32]  Elliott H. Lieb,et al.  A General Rearrangement Inequality for Multiple Integrals , .