In spaces of nonpositive curvature the existence of isometrically embedded
flat (hyper)planes is often granted by apparently weaker conditions on large
scales. We show that some such results remain valid for metric spaces with
non-unique geodesic segments under suitable convexity assumptions on the
distance function along distinguished geodesics. The discussion includes, among
other things, the Flat Torus Theorem and Gromov's hyperbolicity criterion
referring to embedded planes. This generalizes results of Bowditch for Busemann
spaces.Comment: Final version, to appear in Analysis and Geometry in Metric Spaces
(AGMS
