Sectional curvature-type conditions on metric spaces

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincar\'e condition and the measure contraction property. Using a comparison geometry variant for general lower curvature bounds $k\in\mathbb{R}$, a Bonnet-Myers theorem can be proven for spaces with lower curvature bound $k>0$. In the second part the notion of uniform smoothness known from the theory of Banach spaces is applied to metric spaces. It is shown that Busemann functions are (quasi-)convex. This implies the existence of a weak soul. In the end properties are developed to further dissect the soul.