A Generalization of Blaschke's Convergence Theorem in Metric Spaces

A metric space $(X, d)$ together with a set-valued mapping $G: X\times X\to 2^X$ is said to be a generalized segment space $(X, d, G)$ if $G(x, y)\not=\emptyset$ for all $x, y\in X$ and for any sequences $x_n\to x$ and $y_n\to y$ in $X$, $d_H\big(G(x_n, y_n), G(x, y) \big) \to 0$ as $n\to \infty$, where $d_H$ is the Hausdorff distance. Normed linear spaces, nonempty convex sets, and proper uniquely geodesic spaces, etc are generalized segment spaces for suitable $G$. A subset $A$ of $X$ is called \emph{$G$-type convex} if $G(x, y)\subset A$ whenever $x, y\in A$. We prove a generalization of Blaschke's convergence theorem for metric spaces: if $(X, d, G)$ is a proper generalized segment space, then every uniformly bounded sequence of nonempty $G$-type convex subsets of $X$ contains a subsequence which converges to some nonempty compact $G$-type convex subset in $X$.