Geodesic bicombings on some hyperspaces

Abstract

We show that if $(X,d)$ is a metric space which admits a consistent convex geodesic bicombing, then we can construct a conical bicombing on $CB(X)$, the hyperspace of nonempty, closed, bounded, and convex subsets of $X$ (with the Hausdorff metric). If $X$ is a normed space or an $\mathbb{R}$-tree, this same method produces a consistent convex bicombing on $CB(X)$. We follow this by examining a geodesic bicombing on the nonempty compact subsets of $X$, assuming $X$ is a proper metric space.