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 \(\mathcal {J}\subset BCl(X)\) $\mathcal{J}\subset BCl(X)$ that (i) consists of precompact subsets of X, (ii) contains the singletons \(\{x\}\) $\{x\}$ for every \(x\in X\) $x\in X$ , and (iii) satisfies \(BCl(C)\subset \mathcal {J}\) $BCl(C)\subset \mathcal{J}$ for every \(C\in \mathcal {J}\) $C\in \mathcal{J}$ . When X is geodesic, we investigate quasiconvexity of \(\mathcal {J}\) $\mathcal{J}$ for some instances of \(\mathcal {J}\) $\mathcal{J}$ , especially when \(\mathcal {J}\) $\mathcal{J}$ consists of finite subsets of X.