In this talk I want to show that Busemann’s approach to sectional curvature already leads to strong results in the metric setting. More precisely, Busemann's non-positive curvature in combination with the Affine Rigidity Theorem gives Flat Rectangle and Flat Triangle Theorems. Furthermore, in those spaces a contraction of the projection onto convex sets is equivalent to symmetry of the orthogonality relation which is known to hold only for Riemannian manifolds. This shows that one cannot expect such a property to hold in general Busemann convex metric spaces.
On the other hand, a non-negative curvature in the sense of Busemann gives a bi-Lipschitz splitting theorem and shows that spaces with non-trivial Hausdorff measure satisfy the measure contraction property and Poincare inequalities.  

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