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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.
About the author
Mikhail I. Ostrovskii, St. John's University, Queens, USA.
Report
"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
