Metric Spaces, Convexity and Nonpositive Curvature

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European Mathematical Society, Jan 1, 2005 - Mathematics - 287 pages
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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 the theory of geodesics in metric spaces, 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 from classical hyperbolic geometry and from the theory of Teichmuller spaces.
The book is useful for students and researchers in geometry, topology and analysis.
 

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Contents

Some historical markers
1
Lengths of paths in metric spaces
10
Length spaces and geodesic spaces
34
Maps between metric spaces
79
Distances
103
Convexity in vector spaces
127
Convex functions
159
Strictly convex normed vector spaces
178
Busemann spaces
187
Locally convex spaces
210
Asymptotic rays and the visual boundary
229
Isometrics
241
Busemann functions corays and horospheres
261
References
275
Index
283
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