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Metric Embeddings : Bilipschitz and Coarse Embeddings into Banach Spaces.
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....
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Berlin :
De Gruyter,
2013.
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| Colección: | De Gruyter studies in mathematics.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.
- 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.
- 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.
- 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.