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COPYRIGHT 2006 Hindawi Publishing Corp.
Let X be a complete CAT(0) space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if C is a closed subset of X, then the set of points of X which have a unique nearest point in C is [G.sub.[delta] and of the second Baire category in X. If, in addition, C is bounded, then the set of points of X which have a unique farthest point in C is dense in X. A proximity result for set-valued mappings is also included.
1. Introduction
This paper is primarily motivated by a recent paper of Zamfirescu [17], in which it is shown that for any compact set C in a complete length space X without bifurcating geodesics, the nearest point projection [P.sub.C] of X onto C is properly single valued at most points of X, that is, on a set of second Baire category.We show here that the same is true for a closed subset of a complete CAT(0) space X, provided X has the geodesic extension property and has Alexandrov curvature bounded below. We also show that if C is bounded and closed, then the set of points of X which have a unique farthest point in C is dense in X. These are extensions pioneering results, see Edelstein [7] and Steickin [16]. Also see [15] for other generic results.
A metric space is a CAT(0) space (the term is due to Gromov---see, e.g., [1, page 159]) if it is geodesically connected, and if every geodesic triangle in X is at least as "thin" as its comparison triangle in the Euclidean plane. Precise definitions are given below. For a detailed discussion of the properties of such spaces, see Bridson and Haefliger [1] or Burago et al. [3]. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include the classical hyperbolic spaces, Euclidean buildings (see [2]), the complex Hilbert ball with the hyperbolicmetric (see [9]; also [14, inequality (4.3)] and subsequent comments), and many others.
2. Preliminaries
Let (X,d) be a metric space. A geodesic path joining x [member of] X to y [member of] X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] [subset] R to X such that c(0) = x, c(l) = y, and d(c(t), c(t')) = |t -t'| for all t, t' [member of] [0, l]. In particular, c is an isometry and d(x, y) = l. The image [alpha] of c is called a geodesic (or metric) segment joining x and y. When unique, this geodesic is denoted [x, y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y [member of] X. A subset Y [??] X is said to be convex if Y includes every geodesic segment joining any two of its points.
For complete details and further discussion, see, for example, [1] or [3].
For [KAPPA] [member of] (-[infinity],0], let [M.sup.2.sub.k] denote the classical surface of curvature [KAPPA]. Thus [M.sup.2.sub.k] is just the Euclidean plane [E.sup.2] if [KAPPA] = 0, and [M.sup.2.sub.k] is obtained from the classical hyperbolic plane by multiplying the distance function by 1/[square root of-[KAPPA]] if [KAPPA] < 0.
A geodesic triangle [DELTA]([x.sub.1], [x.sub.2], [x.sub.3]) in a geodesic metric space (X,d) consists of three points in X (the vertices of [DELTA]) and a geodesic segment...
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