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Metric Spaces, Convexity and Nonpositive Curvature: Second Edition
Athanase Papadopoulos, Université de Strasbourg, France
A publication of the European Mathematical Society.
IRMA Lectures in Mathematics and Theoretical Physics
2013; 320 pp; softcover
Volume: 6
ISBN-10: 3-03719-132-5
ISBN-13: 978-3-03719-132-3
List Price: US$58
Member Price: US$46.40
Order Code: EMSILMTP/6.R
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This book is about metric spaces of nonpositive curvature in the sense of Busemann, that is, metric spaces whose distance function satisfies a convexity condition. The book also contains a systematic introduction to metric geometry, as well as a detailed presentation of some facets of convexity theory that are useful in the study of nonpositive curvature.

The concepts and the techniques are illustrated by many examples, in particular from hyperbolic geometry, Hilbert geometry and Teichmüller theory.

For the second edition, some corrections and a few additions have been made, and the bibliography has been updated.

A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society.


Graduate students and research mathematicians interested in geometry, topology, and analysis.

Table of Contents

  • Introduction: Some historical markers
  • Lengths of paths in metric spaces
  • Length spaces and geodesic spaces
  • Maps between metric spaces
  • Distances
  • Convexity in vector spaces
  • Convex functions
  • Strictly convex normed vector spaces
  • Busemann spaces
  • Locally convex spaces
  • Asymptotic rays and the visual boundary
  • Isometries
  • Busemann functions, co-rays and horospheres
  • Bibliography
  • Index
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