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El. knyga: Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2014-2016

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  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2169
  • Išleidimo metai: 17-Apr-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319452821
Kitos knygos pagal šią temą:
Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2014-2016Didesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2169
  • Išleidimo metai: 17-Apr-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319452821
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As in the previous Seminar Notes, the current volume reflects general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. A classical theme in the Local Theory of Banach Spaces which is well represented in this volume is the identification of lower-dimensional structures in high-dimensional objects. More recent applications of high-dimensionality are manifested by contributions in Random Matrix Theory, Concentration of Measure and Empirical Processes. Naturally, the Gaussian measure plays a central role in many of these topics, and is also studied in this volume; in particular, the recent breakthrough proof of the Gaussian Correlation Conjecture is revisited. The interplay of the theory with Harmonic and Spectral Analysis is also well apparent in several contributions. The classical relation to both the primal and dual Brunn-Minkowski theories is also well represented, and related algebraic structures pertaining to valuations and valent functions are  discussed. All contributions are original research papers and were subject to the usual refereeing standards.

Recenzijos

 

On Repeated Sequential Closures of Constructive Functions in Valuations
1(14)
Semyon Alesker
1 Main Results
1(3)
2 Proofs
4(11)
References
14(1)
Orbit Point of View on Some Results of Asymptotic Theory; Orbit Type and Cotype
15(10)
Limor Ben-Efraim
Vitali Milman
Alexander Segal
1 Introduction
15(3)
2 Proof of the Extended Kwapien Theorem
18(7)
References
23(2)
Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions
25(30)
Sergey G. Bobkov
Piotr Nayar
Prasad Tetali
1 Introduction
25(4)
2 Bounds on ψα-Norms for Restricted Measures
29(3)
3 Proof of Theorem 1.1: Transport-Entropy Formulation
32(6)
4 Proof of Theorem 1.3: Spectral Gap
38(2)
5 Examples
40(4)
6 Deviations for Non-Lipschitz Functions
44(3)
7 Optimality
47(8)
Appendix
49(3)
References
52(3)
On Random Walks in Large Compact Lie Groups
55(10)
Jean Bourgain
1 Introduction
55(2)
2 Some Preliminary Comments
57(1)
3 Proof of Proposition 1
58(3)
4 Proof of Proposition 2
61(1)
5 Variants
62(3)
References
63(2)
On a Problem of Farrell and Vershynin in Random Matrix Theory
65(6)
Jean Bourgain
1 Introduction
65(1)
2 Proof of the Proposition
66(5)
References
69(2)
Valuations on the Space of Quasi-Concave Functions
71(36)
Andrea Colesanti
Nico Lombardi
1 Introduction
71(4)
2 Notations and Preliminaries
75(3)
2.1 Convex Bodies
75(3)
3 Quasi-Concave Functions
78(5)
3.1 The Space CN
78(2)
3.2 Operations with Quasi-Concave Functions
80(1)
3.3 Three Technical Lemmas
80(3)
4 Valuations
83(1)
4.1 A Brief Discussion on the Choice of the Topology in CN
83(1)
5 Integral Valuations
84(10)
5.1 Continuous Integral Valuations
84(4)
5.2 Monotone (and Continuous) Integral Valuations
88(4)
5.3 The Connection Between the Two Types of Integral Valuations
92(1)
5.4 The Case k = N
93(1)
6 Simple Valuations
94(6)
6.1 Simple Functions
95(1)
6.2 Characterization of Simple Valuations
96(4)
7 Proof of Theorem 1.1
100(2)
8 Monotone Valuations
102(5)
8.1 Identification of the Measures vk, k = 0,..., N
102(1)
8.2 The Case of Simple Functions
103(1)
8.3 Proof of Theorem 1.3
104(1)
References
104(3)
An Inequality for Moments of Log-Concave Functions on Gaussian Random Vectors
107(16)
Nikos Dafnis
Grigoris Paouris
1 Introduction and Main Results
107(2)
2 Proof of the Main Result
109(10)
2.1 Decomposing the Identity
110(4)
2.2 Proof of Theorem 1.1
114(5)
3 Reverse Logarithmic Sobolev Inequality
119(4)
References
121(2)
(s,p)-Valent Functions
123(14)
Omer Friedland
Yosef Yomdin
1 Introduction
123(2)
2 Taylor Domination, Bounded Recurrences
125(2)
3 Distortion Theorem
127(1)
4 Complex Polynomials
128(6)
4.1 Geometric and Analytic Properties of the Invariant cd.α
131(2)
4.2 An Example
133(1)
5 Remez Inequality
134(3)
References
135(2)
A Remark on Projections of the Rotated Cube to Complex Lines
137(14)
Efim D. Gluskin
Yaron Ostrover
1 Introduction and Result
137(3)
2 Preliminaries
140(1)
3 Proof of the Main Theorem
141(10)
Appendix
144(4)
References
148(3)
On the Expectation of Operator Norms of Random Matrices
151(12)
Olivier Guedon
Aicke Hinrichs
Alexander E. Litvak
Joscha Prochno
1 Introduction and Main Results
151(3)
2 Notation and Preliminaries
154(3)
3 Proof of the Main Result
157(3)
4 Concluding Remarks
160(3)
References
161(2)
The Restricted Isometry Property of Subsampled Fourier Matrices
163(18)
Ishay Haviv
Oded Regev
1 Introduction
163(4)
1.1 Our Contribution
165(1)
1.2 Related Literature
165(1)
1.3 Proof Overview
166(1)
2 Preliminaries
167(1)
3 The Simpler Analysis
168(5)
3.1 The Restricted Isometry Property
173(1)
4 The Improved Analysis
173(8)
4.1 The Restricted Isometry Property
177(1)
References
178(3)
Upper Bound for the Dvoretzky Dimension in Milman-Schechtman Theorem
181(6)
Han Huang
Feng Wei
1 Introduction
181(2)
2 Proof of Theorem B
183(4)
References
186(1)
Super-Gaussian Directions of Random Vectors
187(26)
Bo'az Klartag
1 Introduction
187(3)
2 Proof Strategy
190(2)
3 Central Limit Regime
192(6)
4 Geometry of the High-Dimensional Sphere
198(6)
5 Proof of the Main Proposition
204(2)
6 Angularly-Isotropic Position
206(7)
References
211(2)
A Remark on Measures of Sections of Lp-balls
213(8)
Alexander Koldobsky
Alain Pajor
1 Introduction
213(2)
2 Slicing Inequalities
215(6)
References
219(2)
Sharp Poincare-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets
221(14)
Alexander V. Kolesnikov
Emanuel Milman
1 Introduction
221(5)
1.1 Brunn--Minkowski Inequality
224(1)
1.2 Ehrhard Inequality
225(1)
1.3 Comparison with Previous Results
225(1)
2 Proof of Theorem 1.1
226(2)
3 Neumann-to-Dirichlet Operator
228(3)
4 Concluding Remarks
231(4)
4.1 Refined Version
231(1)
4.2 Mean-Curvature Inequality Implies Isoperimetric Inequality
232(1)
4.3 Ehrhard's Inequality is False for CD(1, ∞) Measures
233(1)
4.4 Dual Inequality for Mean-Convex Domains
233(1)
References
234(1)
Rigidity of the Chain Rule and Nearly Submultiplicative Functions
235(30)
Hermann Konig
Vitali Milman
1 Introduction and Results
236(6)
2 Proof of Theorems 1 and 2
242(8)
3 Further Results on Submultiplicativity
250(5)
4 Proof of the Rigidity and the Stability of the Chain Rule
255(10)
References
264(1)
Royen's Proof of the Gaussian Correlation Inequality
265(12)
Rafal Latala
Dariusz Matlak
1 Introduction
265(2)
2 Proof of Theorem 2
267(2)
3 Auxiliary Lemmas
269(8)
References
274(3)
A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets
277(24)
Christopher Liaw
Abbas Mehrabian
Yaniv Plan
Roman Vershynin
1 Introduction
277(4)
2 Applications
281(8)
2.1 Singular Values of Random Matrices
282(1)
2.2 Johnson-Lindenstrauss Lemma
282(1)
2.3 Gordon's Escape Theorem
283(1)
2.4 Sections of Sets by Random Subspaces: The M* Theorem
284(1)
2.5 The Size of Random Linear Images of Sets
284(1)
2.6 Signal Recovery from the Constrained Linear Model
285(2)
2.7 Model Selection for Constrained Linear Models
287(2)
3 Comparison with Known Results
289(1)
4 Preliminaries
290(2)
4.1 Majorizing Measure Theorem, and Deduction of Theorems 1 and 3
290(2)
4.2 Sub-exponential Random Variables, and Bernstein's Inequality
292(1)
5 Proof of Theorem 2
292(3)
6 Proof of Theorem 4
295(2)
7 Further Thoughts
297(4)
References
298(3)
On Multiplier Processes Under Weak Moment Assumptions
301(18)
Shahar Mendelson
1 Introduction
301(7)
2 Proof of Theorem 1.6
308(7)
3 Applications in Sparse Recovery
315(4)
References
318(1)
Characterizing the Radial Sum for Star Bodies
319(12)
Vitali Milman
Liran Rotem
1 Introduction
319(4)
2 The Main Lemma
323(2)
3 Proving the Main Theorems
325(2)
4 Polynomiality of Volume
327(4)
References
329(2)
On Mimicking Rademacher Sums in Tail Spaces
331(8)
Krzysztof Oleszkiewicz
1 Introduction
331(1)
2 Main Result
332(4)
3 Gaussian Counterpart
336(3)
Stability for Borell-Brascamp-Lieb Inequalities
339(22)
Andrea Rossi
Paolo Salani
1 Introduction
339(3)
2 Preliminaries
342(9)
2.1 Notation
342(1)
2.2 About the Brunn-Minkowski Inequality
343(2)
2.3 The Equivalence Between BBL and BM Inequalities
345(6)
3 The Proof of Theorem 1.3
351(4)
4 A Generalization to the Case s Positive Rational
355(4)
5 A Stability for s ≥ 0
359(2)
Appendix 361(1)
References 362
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