Metric Spaces, Convexity and Nonpositive Curvature (Second edition)

Athanase Papadopoulos (1)

1 IRMA - Institut de Recherche Mathématique Avancée

This is the second edition of a book which appeared in 2005. The new edition is an expanded and revised version. The book is about metric spaces of nonpositive curvature in the sense of Busemann, that is, metric spaces whose distance function is convex. We have also included a systematic introduction to the theory of geodesics and related matters in metric spaces, as well as a detailed presentation of a few facets of convexity theory that are useful in the study of nonpositive curvature. The exposition starts from first principles and we give full proofs. Examples and applications are spread throughout the book, and they come from hyperbolic geometry, from the theory of Teichmüller spaces and from Hilbert geometry. At the end of each chapter there are historical notes and other notes on further developments.

Mots clés: Convexity; metric space; metric geometry; nonpositive curvature; Busemann geometry; Alexandrov space; global methods; isometries; Busemann space; locally convex space; visual boundary; Busmeann function; horosphere; Minkowski geometry; Menger convexity; Hausdorff metric; hyperbolic geometry; Teichmüller space; Hilbert geometry.; Hilbert geometry

Domaines: Géométrie métrique [math.MG]; Topologie géométrique [math.GT]

Athanase Papadopoulos. Metric Spaces, Convexity and Nonpositive Curvature (Second edition). European Mathematical Society. European Mathematical Society, 6, 320 p., 2014, IRMA Lectures in Mathematics and Theoretical Physics Vol. 6, C. Kassel et V. Turaev, 978-3-03719-132-3. ⟨10.4171/132⟩. ⟨hal-00943832⟩