Hyperconvexity of ℝ-trees

W. Kirk

Fundamenta Mathematicae (1998)

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

Abstract

top
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

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.