Geometric Aspects of Functional Analysis

This is the sixth published volume of the Israel Seminar on Geometric Aspects of Functional Analysis. The previous volumes are 1983-84 published privately by Tel Aviv University 1985-86 Springer Lecture Notes, Vol. 1267 1986-87 Springer Lecture Notes, Vol. 1317 1987-88 Springer Lecture Notes, Vol. 1376 1989-90 Springer Lecture Notes, Vol. 1469 As in the previous vC!lumes the central subject of -this volume is Banach space theory in its various aspects. In view of the spectacular development in infinite-dimensional Banach space theory in recent years (like the solution of the hyperplane problem, the unconditional basic sequence problem and the distortion problem in Hilbert space) it is quite natural that the present volume contains substantially more contributions in this direction than the previous volumes. This volume also contains many important contributions in the "traditional directions" of this seminar such as probabilistic methods in functional analysis, non-linear theory, harmonic analysis and especially the local theory of Banach spaces and its connection to classical convexity theory in IRn. The papers in this volume are original research papers and include an invited survey by Alexander Olevskii of Kolmogorov's work on Fourier analysis (which was presented at a special meeting on the occasion of the 90th birthday of A. N. Kol­ mogorov). We are very grateful to Mrs. M. Hercberg for her generous help in many directions, which made the publication of this volume possible. Joram Lindenstrauss, Vitali Milman 1992-1994 Operator Theory: Advances and Applications, Vol.

?2-estimate for the euclidean norm on a convex body in isotropic position.- References.- Embedding ?n?-Cubes in low dimensional Schatten classes.- References.- Products of unconditional bodies.- 0 Introduction.- 1 The general Lozanovskii problem for products of unconditional bodies.- 2 Volumes of products of unconditional bodies.- References.- Remarks on Halasz-Montgomery type inequalities.- 1 Introduction.- 2 Proof of Proposition 1.- 3 Proof of Proposition 2.- 4 Zero-density estimates.- References.- Estimates for cone multipliers.- 0 Summary.- 1 L4-estimates.- 2 Kakeya type structures.- 3 A first L2-estimate.- 4 Fourier transform of measures on a cone.- 5 Application to cone multipliers.- References.- Remarks on Bourgain’s problem on slicing of convex bodies.- References.- A note on the Banach-Mazur distance to the cube.- 1 Introduction.- 2 Proof of the Proposition.- 3 Remark.- References.- Projection functions on higher rank Grassmannians.- 1 Introduction.- 2 Projection functions and surface area measures.- 3 The sizes of projection classes.- 4 Radon transforms and projection functions.- References.- On the volume of unions and intersections of balls in Euclidean space.- 1 Introduction.- 2 Volume of flowers in Sn?1 and ?n.- 3 Extension to special cases of N caps in Sn?1.- References.- Uniform non-equivalence between Euclidean and hyperbolic spaces.- 1 Introduction.- 2 Necessary definitions.- 3 The big spheres tangency.- 4 One negative result.- 5 The results.- 6 The proofs.- References.- A hereditarily indecomposable space with an asymptotic unconditional basis.- 1 Introduction.- 2 Some definitions and basic lemmas.- 3 The definition of the space and some of its properties.- 4 Proof of the main result.- References.- Proportional subspaces of spaces with unconditional basis have good volume properties.- 1 Introduction.- 2 Proofs.- References.- A remark about distortion.- References.- Symmetric distortion in ?2.- 1 Symmetric ABS in ?2.- 2 The ?r case.- References.- Asymptotic infinite-dimensional theory of Banach spaces.- 1 Asymptotic and permissible spaces.- 2 Asymptotic versions.- 3 Uniqueness of the asymptotic-?p structure.- 4 Duality of asymptotic-?p spaces.- 5 Complemented permissible subspaces.- References.- On the richness of the set of p’s in Krivine’s theorem.- 1 A space with no spreading model containing c0 or ?p.- 2 A space with a large nonshrinkable Krivine-p-set.- References.- Kolmogorov’s theorems in Fourier analysis.- 0 Introduction.- 1 Kolmogorov’s example of divergent Fourier Series.- 2 Kolmogorov’s weak type inequality.- 3 Kolmogorov’s rearrangement theorem.- References.- Two unexpected examples concerning differentiability of Lipschitz functions on Banach spaces.- 1 Incompatibility of Gâteaux and Fréchet differentiability results.- 2 Strange difference between Fréchet differentiability of Lipschitz functions and of Lipschitz mappings.- References.- Determinant inequalites with applications to isoperimetric inequalities.- 1 Introduction.- 2 Determinant estimates.- 3 Infinite determinants.- 4 Isoperimetric inequalities for simplices.- References.- Approximate John’s decompositions.- References.- Two remarks on 1-unconditional basic sequences in Lp, 3 ? p < ?.- References.- A concentration inequality for harmonic measures on the sphere.- 1 Introduction and notation.- 2 The concentration inequality.- 3 Some corollaries of Theorem 2.1.- 4 Exit times for convex symmetric bodies.- 5 Appendix.- References.- A concentration of measure phenomenon on uniformly convex bodies.- 1 Maurey’s proof.- 2 Uniform convex spaces.- 3 An estimate for the floating body of Bdp.- References.- Embedding of ??k and a theorem of Alon and Milman.- References.- Are all sets of positive measure essentially convex?.- 1 Introduction.- 2 Gauss space.- 3 Some aspects of the solid case.- 4 Sets of sequences.- References.- Embedding subspaces of Lp in ?Np.- 1 Introduction.- 2 The iteration method and the random choice.- 3 Tree extraction.- 4 Entropy estimates.- 5 Main construction.- References.- Distortions on Schatten classes Cp.- 1 Preliminary remarks.- 2 Asymptotic sets in Cp.- References.- GAFA Seminar ― List of Talks.