On Quasiconvexity of Precompact-Subset Spaces

Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$ as a metric space with respect to Hausdorff distance. We study both characterization and representation of Lipschitz paths in $BCl(X)$ in terms of Lipschitz paths in $X$ and in the completion of $X$. We show that a full characterization and representation is possible in any subspace $\mathcal{J}\subset BCl(X)$ that (i) consists of precompact subsets of $X$, (ii) contains the singletons $\{x\}$ for every $x\in X$, and (iii) satisfies $BCl(C)\subset \mathcal{J}$ for every $C\in \mathcal{J}$. When $X$ is geodesic, we investigate quasiconvexity of $\mathcal{J}$ for some instances of $\mathcal{J}$, especially when $\mathcal{J}$ consists of finite subsets of $X$.