A Generalization of Blaschke's Convergence Theorem in Metric Spaces

Journal of Convex Analysis, 20(4) (2013), 1013-1024
ftp://file.viasm.org/Hotro/ChuongtrinhToan/Thuongcongtrinh/ThuongCongTrinh-2013/2013_Bai_160.pdf

A metric space (X, d) together with a set-valued mapping G : X × X ? 2X is said to be a generalized segment space (X, d,G) if G(x, y) 6= ? for all x, y ? X and for any sequences xn ? x and yn ? y in X, dH (G(xn, yn),G(x, y)) ? 0 as n ? ?, where dH 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 G-type convex if
G(x, y) ? A whenever x, y ? 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.