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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
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