Convexities of metric spaces

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Uniform convexity and smoothness, and their applications in Finsler geometry (2009, background): “We close the section with a remark on another type of convexity studied in [19] (see also [10]).”
Barycenters in Alexandrov spaces of curvature bounded below (2012, background): “This condition is regarded as a generalization of the 2-uniform convexity in Banach space theory, see [Oh2, x5] (and [ Oh1 ], [Oh4] as well) for more discussion.”
Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space (2019, background): “nder a uniform positive lower bound on the function (x,b) → kb b⋆(x). Such a condition is closely linked to the better know notion of k-convexity which is connected to positive curvature upper bounds [Oht07]. In fact, it is not hard to check that k min-convexity of the support of P for some k min &g”
Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics (2018, background): “Definition 9 is a slight generalisation of the definition of a k-convex set introduced by Ohta (2007). A k-convex set B, in the sense of Ohta (2007), is a (k, 2)-convex set in the sense of Definition 9. Hence, we simply give additional freedom through the introduction of parameter α ≥ 0. Note that, ”