Lipschitz extensions from spaces of nonnegative curvature into CAT(1) spaces

We prove that complete CAT () spaces of sufficiently small radii possess metric cotype 2 and metric Markov cotype 2. This generalizes the previously known result for complete CAT (0) spaces. The generalization involves extending the variance inequality known for barycenters in CAT (0) spaces to an inequality analogous to one for 2-uniformly convex Banach spaces, and demonstrating that the barycenter map on such spaces is Lipschitz continuous on the corresponding Wasserstein 2 space. By utilizing the generalized Ball extension theorem by Mendel and Naor, we obtain an extension result for Lipschitz maps from Alexandrov spaces of nonnegative curvature into CAT () spaces whose image is contained in a subspace of sufficiently small radius, thereby weakening the curvature assumption in the well-known Lipschitz extension theorem for Alexandrov spaces by Lang and Schr\"oder. As an additional application, we obtain that ₚ spaces for p > 2 cannot be uniformly embedded into any CAT () space with sufficiently small diameter.