Best Approximation in Metric Spaces

A metric space (X, d) is called an M-space if for every x and y in X and for every r 6 [0, A] we have B[x, r] Cl B[y, A — r] = {2} for some z € X, where A = d(x, y). It is the object of this paper to study M-spaces in terms of proximinality properties of certain sets. 0. Introduction. Let (X, d) be a metric space, and G be a closed subset of X. For x E X, let p(x,G) = inf{d(x, y) : y E G}. If the infimum is attained for all x E X, then G is called proximinal in X. The problem of priximinality of subsets in normed spaces has been studied by many authors. We refer mainly to the encyclopedia of Singer [10], and other references cited there, where the problem is treated in detail. Singer suggested the problem of proximinality in the so-called convex metric spaces. In metric linear spaces, many results on the proximinality problem were obtained in [1, 2] and other papers cited in [2]. Some results on proximinality in metric (but necessarity linear) spaces appeared in [6 and 7]. In [7], Busemann considered externally convex metric spaces (X, d) [4, p. 51], which satisfy the following conditions: (i) For any x,y,E X there exists z E X, z & {x, y} such that d(x, z) + d(y, z) = d(x,y) (ii) If x,y,zi,z2 E X such that d(x,y) + d(y,zi) = d(x,zi), d(x,y) + d(yz2) = d(x,z2) and d(y,zi) = d(y,z2), then zx = z2. (iii) Every bounded infinite set in (X, d) has an accumulation point. Busemann proved that the balls in (X, d) are convex if and only if they are Chebyshev. In this paper we prove results which relate proximinality of sets to the metric structure in the space. We prove among other things that Busemann's result is true for a larger class of metric spaces. More results on convexity and Chebyshevity of sets are presented. In §1, we define and characterize M-spaces. In §2, we study the convexity of balls in relation to prominality of convex sets in M-spaces. Some results on proximinality in strictly convex metric spaces are obtained. 1. M-spaces. Let (X, d) be a metric space. For x E X, r > 0, we let B(x,r) = {y E X: d(x,y) < r), B\x,r] = {y E X: d(x,y) < r} and S(x,r) = {y E X: d(x,y) = r}. Following Menger [11], we call (X,d) a convex metric space if B[x, ri ] fl B[y, r2\ ^ 0 whenever rx +r2> d(x, y). Some of the properties of these Received by the editors April 2, 1986 and, in revised form, October 24, 1986 and March 2, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 51K05, 51K99.