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The STUDIA UNIVERSITATIS BABEŞ-BOLYAI issue article summary
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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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