Abstract

A metric space (X,d)(X, d) together with a set-valued mapping G:X×X→2XG: X\times X\to 2^X is said to be a generalized segment space (X,d,G)(X, d, G) if G(x,y)≠∅G(x, y)\not=\emptyset for all x,y∈Xx, y\in X and for any sequences xn→xx_n\to x and yn→yy_n\to y in XX, dH(G(xn,yn),G(x,y))→0d_H\big(G(x_n, y_n), G(x, y) \big) \to 0 as n→∞n\to \infty, where dHd_H is the Hausdorff distance. Normed linear spaces, nonempty convex sets, and proper uniquely geodesic spaces, etc are generalized segment spaces for suitable GG. A subset AA of XX is called GG-type convex if G(x,y)⊂AG(x, y)\subset A whenever x,y∈Ax, y\in A. We prove a generalization of Blaschke's convergence theorem for metric spaces: if (X,d,G)(X, d, G) is a proper generalized segment space, then every uniformly bounded sequence of nonempty GG-type convex subsets of XX contains a subsequence which converges to some nonempty compact GG-type convex subset in XX.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Blaschke's convergence theorem
  • convex sets
  • generalized convexity
  • geodesic convex sets
  • geodesic segments
  • Hausdorff distance
  • uniquely geodesic spaces

Mathematics Subject Classification

52A10, 52B55, 52C45

References

  1. C. D. Aliprantis, O. Burkinshaw: Principles of Real Analysis, 3rd Ed., Academic Press, San Diego (1998).Google Scholar
  2. J. P. Aubin: Applied Abstract Analysis, John Wiley & Sons, New York (1977).Google Scholar
  3. M. R. Bridson, A. Haefliger: Metric Spaces of Non-Positive Curvature, Springer, Berlin (1999).https://doi.org/10.1007/978-3-662-12494-9Google Scholar
  4. N. N. Hai, P. T. An: Blaschke-type theorem and separation of disjoint closed geodesic convex sets, J. Optim. Theor. Appl. 151(3) (2011) 541–551.https://doi.org/10.1007/s10957-011-9868-5Google Scholar
  5. T. Hirose: On the convergence theorem for star-shaped sets in E n, Proc. Japan Acad. 41(3) (1965) 209–211.https://doi.org/10.3792/pja/1195522455Google Scholar
  6. A. Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature, European Mathematical Society, Zürich (2005).https://doi.org/10.4171/010Google Scholar
  7. G. B. Price: On the completeness of a certain metric space with an application to Blaschke’s selection theorem, Bull. Amer. Math. Soc. 46 (1940) 278–280.https://doi.org/10.1090/S0002-9904-1940-07195-2Google Scholar
  8. G. T. Toussaint: Computing geodesic properties inside a simple polygon, Rev. Intell. Artif. 3 (1989) 9–42.Google Scholar
  9. F. A. Valentine: Convex Sets, McGraw-Hill, New York (1964).Google Scholar
  10. R. Webster: Convexity, Oxford University Press, Oxford (1994).https://doi.org/10.1093/oso/9780198531470.001.0001Google Scholar

Suggested citation

N. N. Hai, P. T. An. “A Generalization of Blaschke's Convergence Theorem in Metric Spaces.” Journal of Convex Analysis 20 (2013), No. 4, 1013–1024. https://doi.org/10.68381/jca20055

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.