tailieunhanh - Metric trees, hyperconvex hulls and extensions

In thispaper we examine the relationship between hyperconvex hullsand metric trees. After providing a linking construction for hyperconvex spaces, we show that the four-point property isinherited by the hyperconvex hull, which leadsto the theorem that every complete metric tree is hyperconvex. | Turk J Math 32 (2008) , 219 – 234. ¨ ITAK ˙ c TUB Metric Trees, Hyperconvex Hulls and Extensions A. G. Aksoy and B. Maurizi Abstract In this paper we examine the relationship between hyperconvex hulls and metric trees. After providing a linking construction for hyperconvex spaces, we show that the four-point property is inherited by the hyperconvex hull, which leads to the theorem that every complete metric tree is hyperconvex. We also consider some extension theorems for these spaces. Key Words: Hyperconvex spaces, metric trees, extensions. 1. Introduction The purpose of this paper is to clarify the relationship between metric trees and hy- perconvex metric spaces. We provide a new so-called linking construction of hyperconvex spaces and show that the four-point property of a metric space is inherited by the hyperconvex hull of that space. We prove that all complete metric trees are hyperconvex. This in turn suggests a new approach to the study of extensions of operators. For a metric space (X, d) we use B(x; r) to denote the closed ball centered at x with radius r ≥ 0. Definition A metric space (X, d) is said to be hyperconvex if i I B(xi ; ri ) = φ for every collection B(xi ; ri ) of closed balls in X for which d(xi , xj ) ≤ ri + rj . This notion was first introduced by Aronszajn and Panitchpakdi in [1], where it is shown that a metric space is hyperconvex if and only if it is injective with respect to nonexpansive mappings. Later Isbell [11] showed that every metric space has an AMS Mathematics Subject Classification: 05C12, 54H12, 46M10. 219 AKSOY, MAURIZI injective hull, therefore every metric space is isometric to a subspace of a minimal hyperconvex space. Hyperconvex metric spaces are complete and connected [13]. The simplest examples of hyperconvex spaces are the set of real numbers R, or a finitedimensional real Banach space endowed with the maximum norm. While the Hilbert space l2 fails to be hyperconvex, the spaces L∞ and l∞ are .