Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces

Mikhail I. Ostrovskii

Published by De Gruyter 2013

Metric Embeddings

Bilipschitz and Coarse Embeddings into Banach Spaces

Mikhail I. Ostrovskii

Volume 49 in the series De Gruyter Studies in Mathematics

https://doi.org/10.1515/9783110264012

Overview
Contents

About this book

Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. 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: (1) Embeddability of locally finite metric spaces into Banach spaces is finitely determined; (2) Constructions of embeddings; (3) Distortion in terms of Poincaré inequalities; (4) Constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees; (5) Banach spaces which do not admit coarse embeddings of expanders; (6) Structure of metric spaces which are not coarsely embeddable into a Hilbert space; (7) Applications of Markov chains to embeddability problems; (8) Metric characterizations of properties of Banach spaces; (9) Lipschitz free spaces. Substantial part of the book is devoted to a detailed presentation of relevant results of Banach space theory and graph theory. The final chapter contains a list of open problems. Extensive bibliography is also included. Each chapter, except the open problems chapter, contains exercises and a notes and remarks section containing references, discussion of related results, and suggestions for further reading.

The book will help readers to enter and to work in a very rapidly developing area having many important connections with different parts of mathematics and computer science.

Author / Editor information

Mikhail I. Ostrovskii, St. John's University, Queens, USA.

Reviews

"Ostrovskii's Metric Embeddings: Bilipchitz and Coarse Embeddings into Banach Spaces is a very valuable addition to the literature. It contains an impressive amount of material and is recommended to anyone having some interest in these geometric problems. The area is developing at an extremely fast pace and it is difficult to find in a book format the recent developments; the monograph under review contains some very interesting ones."
Florent B. Baudier and William B. Johnson, Bulletin of the American Mathematical Society

"The author has succeeded in making profound material from both the functional analytic and the discrete side accessible to his readers. All the arguments are exposed in great detail, and the line of reasoning is laid out clearly. The book can be highly recommended to researchers who wish to enter the rapidly developing area of metric embeddings."
Dirk Werner, Zentralblatt für Mathematik

Supplementary Materials

Contents

Topics

Frontmatter Publicly Available Download PDF	i
Preface Requires Authentication Unlicensed Licensed Download PDF	v
Contents Publicly Available Download PDF	vii
Chapter 1. Introduction: examples of metrics, embeddings, and applications Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Related Banach space theory Requires Authentication Unlicensed Licensed Download PDF	34
Chapter 3. Constructions of embeddings Requires Authentication Unlicensed Licensed Download PDF	80
Chapter 4. Obstacles for embeddability: Poincaré inequalities Requires Authentication Unlicensed Licensed Download PDF	105
Chapter 5. Families of expanders and of graphs with large girth Requires Authentication Unlicensed Licensed Download PDF	131
Chapter 6. Banach spaces which do not admit uniformly coarse embeddings of expanders Requires Authentication Unlicensed Licensed Download PDF	191
Chapter 7. Structure properties of spaces which are not coarsely embeddable into a Hilbert space Requires Authentication Unlicensed Licensed Download PDF	218
Chapter 8. Applications of Markov chains to embeddability problems Requires Authentication Unlicensed Licensed Download PDF	228
Chapter 9. Metric characterizations of classes of Banach spaces Requires Authentication Unlicensed Licensed Download PDF	265
Chapter 10. Lipschitz free spaces Requires Authentication Unlicensed Licensed Download PDF	308
Chapter 11. Open problems Requires Authentication Unlicensed Licensed Download PDF	328
Bibliography Requires Authentication Unlicensed Licensed Download PDF	335
Author index Requires Authentication Unlicensed Licensed Download PDF	361
Subject index Requires Authentication Unlicensed Licensed Download PDF	367

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Graduate and PhD Students, Researchers; Academic Libraires
Pages
- Front matter: 12
- Main content: 372
Illustrations
- Tables: 2
Keywords: Embedding of Discrete Metric Spaces; Bilipschitz Embedding; Coarse Embedding; Banach Space Theory; Graph Theory; Functional Analysis

eBook

Published: June 26, 2013
ISBN: 9783110264012

Hardcover

Published: June 18, 2013
ISBN: 9783110263404