Abstract
We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.
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Ohta, Si. Convexities of metric spaces. Geom Dedicata 125, 225–250 (2007). https://doi.org/10.1007/s10711-007-9159-3
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DOI: https://doi.org/10.1007/s10711-007-9159-3