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

Abstract

We prove that complete $\text{CAT}(\kappa)$ spaces of sufficiently small radii possess metric cotype 2 and metric Markov cotype 2. This generalizes the previously known result for complete $\text{CAT}(0)$ spaces. The generalization involves extending the variance inequality known for barycenters in $\text{CAT}(0)$ 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 $\text{CAT}(\kappa)$ 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 $\ell_p$ spaces for $p > 2$ cannot be uniformly embedded into any $\text{CAT}(\kappa)$ space with sufficiently small diameter.