A selection theorem in metric trees

Aksoy, A.; Khamsi, M.

doi:10.1090/S0002-9939-06-08555-8

A selection theorem in metric trees
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by A. G. Aksoy and M. A. Khamsi

Proc. Amer. Math. Soc. 134 (2006), 2957-2966

DOI: https://doi.org/10.1090/S0002-9939-06-08555-8

Published electronically: May 1, 2006

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Abstract:

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping $T^*$ of a metric tree $M$ with convex values has a selection $T: M\rightarrow M$ for which $d(T(x),T(y))\leq d_H(T^*(x),T^*(y))$ for each $x,y \in M$. Here by $d_H$ we mean the Hausdroff distance. Many applications of this result are given.

References

A. G. Aksoy, M. A. Khamsi, Fixed points of uniformly Lipschitzian mappings in metric trees. Preprint.
A. G. Aksoy, B. Maurizi, Metric trees hyperconvex hulls and Extensions. submitted.
N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. MR 84762, DOI 10.2140/pjm.1956.6.405
Jean-Bernard Baillon, Nonexpansive mapping and hyperconvex spaces, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 11–19. MR 956475, DOI 10.1090/conm/072/956475
I. Bartolini, P. Ciaccia, and M. Patella, String matching with metric trees using approximate distance. SPIR, Lecture Notes in Computer Science, Springer Verlag, Vol. 2476 (2002), 271–283.
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
Mladen Bestvina, $\Bbb R$-trees in topology, geometry, and group theory, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 55–91. MR 1886668
Leonard M. Blumenthal, Theory and applications of distance geometry, 2nd ed., Chelsea Publishing Co., New York, 1970. MR 0268781
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Peter Buneman, A note on the metric properties of trees, J. Combinatorial Theory Ser. B 17 (1974), 48–50. MR 363963, DOI 10.1016/0095-8956(74)90047-1
Andreas W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984), no. 3, 321–402. MR 753872, DOI 10.1016/0001-8708(84)90029-X
Andreas Dress, Vincent Moulton, and Werner Terhalle, $T$-theory: an overview, European J. Combin. 17 (1996), no. 2-3, 161–175. Discrete metric spaces (Bielefeld, 1994). MR 1379369, DOI 10.1006/eujc.1996.0015
J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. MR 182949, DOI 10.1007/BF02566944
Mohamed A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996), no. 1, 298–306. MR 1418536, DOI 10.1006/jmaa.1996.0438
Mohamed A. Khamsi and William A. Kirk, An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001. MR 1818603, DOI 10.1002/9781118033074
M. A. Khamsi, W. A. Kirk, and Carlos Martinez Yañez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3275–3283. MR 1777578, DOI 10.1090/S0002-9939-00-05777-4
M. Amine Khamsi, Michael Lin, and Robert Sine, On the fixed points of commuting nonexpansive maps in hyperconvex spaces, J. Math. Anal. Appl. 168 (1992), no. 2, 372–380. MR 1175996, DOI 10.1016/0022-247X(92)90165-A
W. A. Kirk, Hyperconvexity of $\mathbf R$-trees, Fund. Math. 156 (1998), no. 1, 67–72. MR 1610559
W. A. Kirk, Personal communication.
J. C. Mayer, L. K. Mohler, L. G. Oversteegen, and E. D. Tymchatyn, Characterization of separable metric $\textbf {R}$-trees, Proc. Amer. Math. Soc. 115 (1992), no. 1, 257–264. MR 1124147, DOI 10.1090/S0002-9939-1992-1124147-5
John C. Mayer and Lex G. Oversteegen, A topological characterization of $\textbf {R}$-trees, Trans. Amer. Math. Soc. 320 (1990), no. 1, 395–415. MR 961626, DOI 10.1090/S0002-9947-1990-0961626-8
John W. Morgan, $\Lambda$-trees and their applications, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 87–112. MR 1100579, DOI 10.1090/S0273-0979-1992-00237-9
Frank Rimlinger, Free actions on $\textbf {R}$-trees, Trans. Amer. Math. Soc. 332 (1992), no. 1, 313–329. MR 1098433, DOI 10.1090/S0002-9947-1992-1098433-6
Charles Semple and Mike Steel, Phylogenetics, Oxford Lecture Series in Mathematics and its Applications, vol. 24, Oxford University Press, Oxford, 2003. MR 2060009
Robert Sine, Hyperconvexity and nonexpansive multifunctions, Trans. Amer. Math. Soc. 315 (1989), no. 2, 755–767. MR 954603, DOI 10.1090/S0002-9947-1989-0954603-6
J. Tits, A “theorem of Lie-Kolchin” for trees, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 377–388. MR 0578488
M. Zippin, Applications of Michael’s continuous selection theorem to operator extension problems, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1371–1378. MR 1487350, DOI 10.1090/S0002-9939-99-04777-2