The Metric Dimension of Metric Spaces
- PDF / 198,833 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 70 Downloads / 280 Views
The Metric Dimension of Metric Spaces Sheng Bau · Alan F. Beardon
Received: 21 August 2012 / Revised: 15 April 2013 / Accepted: 20 May 2013 / Published online: 20 July 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract Let (X, d) be a metric space. A subset A of X resolves X if each point x in X is uniquely determined by the distances d(x, a), where a ∈ A. The metric dimension of (X, d) is the smallest integer k such that there is a set A of cardinality k that resolves X . Much is known about the metric dimension when X is the vertex set of a graph, but very little seems to be known for a general metric space. Here we provide some basic results for general metric spaces. Keywords
Metric spaces · Metric basis · Dimension · Resolving sets
Mathematics Subject Classification (2000)
51K05 · 51F15 · 05C12
Communicated by Alexander Aptekarev. The work of the S. Bau was supported by Natural Science Foundation of China Grant No. 10971027 at the Center for Discrete Mathematics, Fuzhou University. Present Address: S. Bau University of Witswatersrand, Johannesburg, South Africa e-mail: [email protected] S. Bau School of Mathematics, Inner Mongolia University of Nationalities, Tongliao, China A. F. Beardon (B) African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town, South Africa e-mail: [email protected]; [email protected]
123
296
S. Bau, A. F. Beardon
1 Introduction Let (X, d) be a metric space, A non-empty subset A of X resolves (X, d) if d(x, a) = d(y, a) for all a in A implies x = y, and if this is so we may regard the distances d(x, a), where a ∈ A, as the co-ordinates of x with respect to A. The metric dimension β(X ) of (X, d) is the smallest integer k such that there is a set A of cardinality k that resolves X (and the metric d will always be understood from the context). A set of cardinality β(X ) that resolves X is called a metric basis of X . As X resolves X , every metric space X has a metric dimension which is at most the cardinality |X | of X . The concept of a metric basis of a general metric space first appeared in 1953 in [8], but it attracted little attention until 1975 when it was applied to the set of vertices of a graph [16,29]. Since then, and motivated by problems in graph theory, chemistry, biology, robotics and many other disciplines, much has been published on this topic; see, for example [9,11,12,20,23]. In this paper, we return to the original idea of the metric dimension of a general metric space. Since very little is known about the metric dimension in general metric spaces (apart from graphs with the usual graph metric), it is natural to begin our discussion by studying the simplest case, namely Rn . It is easy to see (and well known) that Euclidean space Rn has metric dimension n + 1; here we prove results about the metric dimension of open sets, convex sets, and sets in affine geometry. We also show that similar results hold for the n-dimensional hyperbolic and spherical spaces (each of constant curvature), and that β(R) = 3 for each Riemann
Data Loading...
