Metric Spaces, Convexity and Nonpositive Curvature

affine segment affinely reparametrized geodesic arbitrary points ball B(x Busemann space Chapter closed ball co-ray completes the proof consider contained converges convex functions convex subset Corollary covering map defined definition distance function distance-preserving endpoints equal equipped Euclidean space examples finite fixed point geodesic lines geodesic metric space geodesic path geodesic segment joining Hausdorff distance homeomorphism horospheres hyperbolic space implies infimum integer interval K-Lipschitz Lemma length metric length space let f let us prove local isometry locally compact locally convex map f map ƒ Menger neighborhood nonempty nonpositive curvature normed vector space notion obtain open ball parametrized by arclength path joining plane positive real number proof of Proposition properties pseudo-metric radius rectifiable paths reparametrized local geodesic respectively Riemannian manifolds sequence space and let strictly convex subdivision surface t₁ Teichmüller space Theorem topology triangle inequality uniquely geodesic space