The study of injective envelopes of metric spaces, also known as metric trees (R-trees or T-theory), has its motivation in many sub-disciplines of mathematics as well as biology/medicine and computer science. Its relationship with biology and medicine stems from the construction of phylogenetic trees [5].Concepts of string matching in computer science is closely related with the structure of metric trees [4]. A metric tree is a metric space such that for every in M there is a unique arc between x and y and this arc is isometric to an interval in R. [3],[2]. In this talk, we examine convexity and compact structures in metric trees and 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. We show that a set valued mapping of a metric tree M with convex values has a selection for which for each . Here by we mean the Hausdroff distance [1]. We will mention some applications to k-set contractions as well as an application of the above selection theorem. Furthermore we define n-widths of a subset A of a metric tree M and show that even in the absence of linear structure the limit of n-widths as is equal to the ball measure of noncompactness.

References

1.A. G.Aksoy, M.A. Khamsi A Selection Theorem in Metric Trees, Proc. Amer. Math. Soc. 134, No.10 (2006), 2957-2966.

2.W. B. Johnson, J. Lindenstrauss and D. Preiss, Lipschitz quotients from metric trees and from Banach spaces containing , J. Funct. Anal. 194 (2002), 332-346.

3.A. W. M. Dress, V. Moulton and W. Terhalle, T-Theory, An overview. European J. Combin. 17 (1996), 161-175.

4.I. Bartolini, P. Ciaccia, and M. Patella, String Matching with metric trees using approximate distance. SPIR, Lecture notes in Computer science, Springer Verlag, 2476, (2002), 271-283.

5.C. Semple, and M. Steel, Phylogenetics. Oxford lecture series in mathematics and its applications, 24, (2003).