## Metric Spaces, Convexity and Nonpositive CurvatureThis 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 |

275 | |

283 | |

### Common terms and phrases

affine segment affinely convex subset affinely reparametrized geodesic arbitrary points asymptotic axial isometry bounded Busemann space Caratheodory Chapter closed ball closed limit co-ray completes the proof consider contained converges convex functions convex subset Corollary covering map defined definition denote distance function distance-preserving endpoints equipped Euclidean space examples fact finite fixed point geodesic lines geodesic metric space geodesic path geodesic segment joining geodesically convex Hadamard Hausdorff distance homeomorphism hyperbolic space implies infimum integer interval Isom(X Kobayashi Lemma length metric length space let f let us prove locally compact Menger neighborhood non-expanding nonempty nonpositive curvature normed vector space notion obtain open ball parametrized by arclength path joining positive real number proof of Proposition properties pseudo-distance pseudo-metric R-tree radius rectifiable paths reparametrized local geodesic respectively Riemannian manifolds satisfying sequence space and let strictly convex suppose surface Teichmiiller space Theorem topology triangle inequality