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Journal of Convex Analysis 20 (2013), No. 4, 1013--1024
Copyright Heldermann Verlag 2013

A Generalization of Blaschke's Convergence Theorem in Metric Spaces

Nguyen Ngoc Hai
Dept. of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam

Phan Thanh An
Center for Mathematics and its Applications, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
and: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, Cau Giay - Hanoi, Vietnam


A metric space $(X, d)$ together with a set-valued mapping $G: X\times X\to 2^X$ is said to be a \emph{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 $X$.

Keywords: Blaschke's convergence theorem, convex sets, generalized convexity, geodesic convex sets, geodesic segments, Hausdorff distance, uniquely geodesic spaces.

MSC: 52A10, 52B55, 52C45

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