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    STUDIA MATHEMATICA - Issue no. 3 / 2007  
  Article:   BOOK REVIEWS.

  Abstract:  Geodesic metric spaces form a class of metric spaces in which convexity of subsets can be defined as well as other related analytic concepts. Buseman spaces are geodesic metric spaces whose length function satisfies a convexity condition. Beside their intrinsic geometric interest, geodesic metric spaces are important for their applications to complex analysis and nonlinear analysis - fixed point theory for nonexpansive mappings, generalized differentiability and optimization. Classical examples of geodesic metrics are the Riemannian metric, the Poincarémetric on the hyperbolic ball Hn, the Carathéodori and the Kobayashi distances for complex manifolds, Thusrston’s metric on complex projective surfaces, the Teichmüller metric and Teichm üller spaces. The book starts with a short historical overview emphasizing some corner points in its development - the pioneering work of J. Hadamard, the contributions of K. Menger, A. Wald, H. Busemann and A. D. Alexandrov. In order to make the book self-contained the author systematically develop in the first two chapters 1. Lengths of paths in metric spaces, 2. Length spaces and geodesic spaces, the basic construction and the properties of length spaces and geodesic spaces, including convexity - geodesic convexity and Menger convexity, this last being defined through the betweeness relation. Chapter 3. Maps between metric spaces, is concerned with Lipschitz maps and fixed points for contractive and for nonexpansive mappings on geodesic spaces. The analog of Hausdorff distance for subsets of a geodesic metric space, called the Busemann-Hausdorff distance, with applications to limits of subsets is considered in Chapter 4. Distances.  
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