Format:
Online-Ressource (XI, 372 S.)
Edition:
Reproduktion 2013
ISBN:
9783110264012
Series Statement:
De Gruyter Studies in Mathematics 49
Content:
Biographical note: Mikhail I. Ostrovskii, St. John's University, Queens,USA.
Content:
Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The book will help readers to enter and to work in this very rapidly developing area having many important connections with different parts of mathematics and computer science. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include embeddability of locally finite metric spaces into Banach spaces is finitely determined, constructions of embeddings, distortion in terms of Poincaré inequalities, constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees, Banach spaces which do not admit coarse embeddings of expanders, structure of metric spaces which are not coarsely embeddable into a Hilbert space, applications of Markov chains to embeddability problem, metric characterizations of properties of Banach spaces, and Lipschitz free spaces.
Note:
Description based upon print version of record
,
Preface; 1 Introduction: examples of metrics, embeddings, and applications; 1.1 Metric spaces: definitions and main examples; 1.2 Types of embeddings: isometric, bilipschitz, coarse, and uniform; 1.2.1 Isometric embeddings; 1.2.2 Bilipschitz embeddings; 1.2.3 Coarse and uniform embeddings; 1.3 Probability theory terminology and notation; 1.4 Applications to the sparsest cut problem; 1.5 Exercises; 1.6 Notes and remarks; 1.6.1 To Section 1.1; 1.6.2 To Section 1.2; 1.6.3 To Section 1.3; 1.6.4 To Section 1.4; 1.6.5 To exercises; 1.7 On applications in topology; 1.8 Hints to exercises
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2 Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Related Banach space theory2.1 Introduction; 2.2 Banach space theory: ultrafilters, ultraproducts, finite representability; 2.2.1 Ultrafilters; 2.2.2 Ultraproducts; 2.2.3 Finite representability; 2.3 Proofs of the main results on relations between embeddability of a locally finite metric space and its finite subsets; 2.3.1 Proof in the bilipschitz case; 2.3.2 Proof in the coarse case; 2.3.3 Remarks on extensions of finite determination results
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2.4 Banach space theory: type and cotype of Banach spaces, Khinchin and Kahane inequalities2.4.1 Rademacher type and cotype; 2.4.2 Kahane-Khinchin inequality; 2.4.3 Characterization of spaces with trivial type or cotype; 2.5 Some corollaries of the theorems on finite determination of embeddability of locally finite metric spaces; 2.6 Exercises; 2.7 Notes and remarks; 2.8 Hints to exercises; 3 Constructions of embeddings; 3.1 Padded decompositions and their applications to constructions of embeddings; 3.2 Padded decompositions of minor-excluded graphs
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3.3 Padded decompositions in terms of ball growth3.4 Gluing single-scale embeddings; 3.5 Exercises; 3.6 Notes and remarks; 3.7 Hints to exercises; 4 Obstacles for embeddability: Poincaré inequalities; 4.1 Definition of Poincaré inequalities for metric spaces; 4.2 Poincaré inequalities for expanders; 4.3 Lp-distortion in terms of constants in Poincaré inequalities; 4.4 Euclidean distortion and positive semidefinite matrices; 4.5 Fourier analytic method of getting Poincaré inequalities; 4.6 Exercises; 4.7 Notes and remarks; 4.8 A bit of history of coarse embeddability; 4.9 Hints to exercises
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5 Families of expanders and of graphs with large girth5.1 Introduction; 5.2 Spectral characterization of expanders; 5.3 Kazhdan's property (T) and expanders; 5.4 Groups with property (T); 5.4.1 Finite generation of SLn(ℤ); 5.4.2 Finite quotients of SLn(ℤ); 5.4.3 Property (T) for groups SLn(ℤ); 5.4.4 Criterion for property (T); 5.5 Zigzag products; 5.6 Graphs with large girth: basic definitions; 5.7 Graph lift constructions and ℓ1-embeddable graphs with large girth; 5.8 Probabilistic proof of existence of expanders; 5.9 Size and diameter of graphs with large girth: basic facts
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5.10 Random constructions of graphs with large girth
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Preface; 1 Introduction: examples of metrics, embeddings, and applications; 1.1 Metric spaces: definitions and main examples; 1.2 Types of embeddings: isometric, bilipschitz, coarse, and uniform; 1.2.1 Isometric embeddings; 1.2.2 Bilipschitz embeddings; 1.2.3 Coarse and uniform embeddings; 1.3 Probability theory terminology and notation; 1.4 Applications to the sparsest cut problem; 1.5 Exercises; 1.6 Notes and remarks; 1.6.1 To Section 1.1; 1.6.2 To Section 1.2; 1.6.3 To Section 1.3; 1.6.4 To Section 1.4; 1.6.5 To exercises; 1.7 On applications in topology; 1.8 Hints to exercises
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2 Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Related Banach space theory2.1 Introduction; 2.2 Banach space theory: ultrafilters, ultraproducts, finite representability; 2.2.1 Ultrafilters; 2.2.2 Ultraproducts; 2.2.3 Finite representability; 2.3 Proofs of the main results on relations between embeddability of a locally finite metric space and its finite subsets; 2.3.1 Proof in the bilipschitz case; 2.3.2 Proof in the coarse case; 2.3.3 Remarks on extensions of finite determination results
,
2.4 Banach space theory: type and cotype of Banach spaces, Khinchin and Kahane inequalities2.4.1 Rademacher type and cotype; 2.4.2 Kahane-Khinchin inequality; 2.4.3 Characterization of spaces with trivial type or cotype; 2.5 Some corollaries of the theorems on finite determination of embeddability of locally finite metric spaces; 2.6 Exercises; 2.7 Notes and remarks; 2.8 Hints to exercises; 3 Constructions of embeddings; 3.1 Padded decompositions and their applications to constructions of embeddings; 3.2 Padded decompositions of minor-excluded graphs
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3.3 Padded decompositions in terms of ball growth3.4 Gluing single-scale embeddings; 3.5 Exercises; 3.6 Notes and remarks; 3.7 Hints to exercises; 4 Obstacles for embeddability: Poincaré inequalities; 4.1 Definition of Poincaré inequalities for metric spaces; 4.2 Poincaré inequalities for expanders; 4.3 Lp-distortion in terms of constants in Poincaré inequalities; 4.4 Euclidean distortion and positive semidefinite matrices; 4.5 Fourier analytic method of getting Poincaré inequalities; 4.6 Exercises; 4.7 Notes and remarks; 4.8 A bit of history of coarse embeddability; 4.9 Hints to exercises
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5 Families of expanders and of graphs with large girth5.1 Introduction; 5.2 Spectral characterization of expanders; 5.3 Kazhdan's property (T) and expanders; 5.4 Groups with property (T); 5.4.1 Finite generation of SLn(ℤ); 5.4.2 Finite quotients of SLn(ℤ); 5.4.3 Property (T) for groups SLn(ℤ); 5.4.4 Criterion for property (T); 5.5 Zigzag products; 5.6 Graphs with large girth: basic definitions; 5.7 Graph lift constructions and ℓ1-embeddable graphs with large girth; 5.8 Probabilistic proof of existence of expanders; 5.9 Size and diameter of graphs with large girth: basic facts
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5.10 Random constructions of graphs with large girth
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In English
Additional Edition:
ISBN 9783110263404
Additional Edition:
ISBN 9783110264012
Additional Edition:
Erscheint auch als Druck-Ausgabe ISBN 978-3-11-026401-2
Additional Edition:
Erscheint auch als Druck-Ausgabe Ostrovskii, Mikhail I. Metric embeddings Berlin : de @Gruyter, 2013 ISBN 9783119166225
Additional Edition:
ISBN 3110263408
Additional Edition:
ISBN 9783110263404
Language:
English
Subjects:
Mathematics
Keywords:
Einbettung
;
Diskreter metrischer Raum
;
Banach-Raum
;
Electronic books
DOI:
10.1515/9783110264012
URL:
Volltext
(lizenzpflichtig)
Author information:
Ostrovskii, Mikhail I.
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