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
nnhai@hcmiu.edu.vn
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
thanhan@math.ist.utl.pt
[Abstract-pdf]
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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