Hyperconvexity of ℝ-trees

W. Kirk

Fundamenta Mathematicae (1998)

  • Volume: 156, Issue: 1, page 67-72
  • ISSN: 0016-2736

Abstract

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It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.

How to cite

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Kirk, W.. "Hyperconvexity of ℝ-trees." Fundamenta Mathematicae 156.1 (1998): 67-72. <http://eudml.org/doc/212261>.

@article{Kirk1998,
abstract = {It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.},
author = {Kirk, W.},
journal = {Fundamenta Mathematicae},
keywords = {hyperconvex metric space; ℝ-tree; fixed point; nonexpansive mapping; complete -tree},
language = {eng},
number = {1},
pages = {67-72},
title = {Hyperconvexity of ℝ-trees},
url = {http://eudml.org/doc/212261},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Kirk, W.
TI - Hyperconvexity of ℝ-trees
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 1
SP - 67
EP - 72
AB - It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.
LA - eng
KW - hyperconvex metric space; ℝ-tree; fixed point; nonexpansive mapping; complete -tree
UR - http://eudml.org/doc/212261
ER -

References

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  1. [1] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439. Zbl0074.17802
  2. [2] J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, in: Fixed Point Theory and its Applications, R. F. Brown (ed.), Contemp. Math. 72, Amer. Math. Soc., Providence, R.I., 1988, 11-19. 
  3. [3] L. M. Blumenthal, Distance Geometry, Oxford Univ. Press, London, 1953. 
  4. [4] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 439-447. Zbl0151.30205
  5. [5] E. Jawhari, D. Misane and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, in: Combinatorics and Ordered Graphs, I. Rival (ed.), Contemp. Math. 57, Amer. Math. Soc., Providence, R.I., 1986, 175-226. Zbl0597.54028
  6. [6] M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723-726. Zbl0671.47052
  7. [7] M. A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996), 298-306. Zbl0869.54045
  8. [8] W. A. Kirk and S. S. Shin, Fixed point theorems in hyperconvex spaces, Houston J. Math. 23 (1997), 175-188. Zbl0957.46033
  9. [9] J. Kulesza and T. C. Lim, On weak compactness and countable weak compactness in fixed point theory, Proc. Amer. Math. Soc. 124 (1996), 3345-3349. Zbl0865.47044
  10. [10] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974. Zbl0285.46024
  11. [11] R. Mańka, Association and fixed points, Fund. Math. 91 (1976), 105-121. Zbl0335.54037
  12. [12] J. W. Morgan, Λ-trees and their applications, Bull. Amer. Math. Soc. 26 (1992), 87-112. Zbl0767.05054
  13. [13] F. Rimlinger, Free actions on ℝ-trees, Trans. Amer. Math. Soc. 332 (1992), 313-329. Zbl0803.20017
  14. [14] R. Sine, On nonlinear contractions in sup norm spaces, Nonlinear Anal. 3 (1979), 885-890. Zbl0423.47035
  15. [15] P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29. Zbl0371.47048
  16. [16] F. Sullivan, Ordering and completeness of metric spaces, Nieuw Arch. Wisk. (3) 29 (1981), 178-193. Zbl0484.54024
  17. [17] J. Tits, A "Theorem of Lie-Kolchin" for trees, in: Contributions to Algebra: a Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977, 377-388. 

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