Metric Spaces, Convexity and Nonpositive Curvature

affinely affinely reparametrized angle arbitrary associated boundary bounded Busemann space called Chapter closed co-ray compact connected consider contained continuous converges convex functions convex subset Corollary covering curvature curve defined definition denote distance distinct element equal equicontinuous equipped equivalent Euclidean examples exists fact Figure finite geodesic path geodesic ray geodesic segment geodesically convex geometry given gives homeomorphism hyperbolic implies induced inequality instance integer interval isometry joining Lemma length space Let X limit locally manifold metric space neighborhood nonpositive normed vector space notion obtain plane positive projection Proof proof of Proposition proper properties Proposition prove radius real number recall relation respectively result satisfying sense sequence shows space and let sphere starting strictly convex Suppose surface Theorem theory topology triangle uniquely geodesic x and y