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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A selection theorem in metric trees


Authors: A. G. Aksoy and M. A. Khamsi
Journal: Proc. Amer. Math. Soc. 134 (2006), 2957-2966
MSC (2000): Primary 47H04, 47H10, 54H25, 47H09
Published electronically: May 1, 2006
MathSciNet review: 2231620
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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.


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Additional Information

A. G. Aksoy
Affiliation: Department of Mathematics, Claremont McKenna College, Claremont, California 91711
Email: asuman.aksoy@claremontmckenna.edu

M. A. Khamsi
Affiliation: Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514
Email: mohamed@math.utep.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08555-8
PII: S 0002-9939(06)08555-8
Keywords: Metric trees, selection theorems, hyperconvex spaces
Received by editor(s): April 27, 2005
Published electronically: May 1, 2006
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society