Discrete Mathematics and Combinatorics Commons^™

Finite Asymptotic Clusters Of Metric Spaces, Viktoriia Bilet, Oleksiy Dovgoshey

Theory and Applications of Graphs

Let (X, d) be an unbounded metric space and let \tilde r=(r_n)_{n\in\mathbb N} be a sequence of positive real numbers tending to infinity. A pretangent space \Omega_{\infty, \tilde r}^{X} to (X, d) at infinity is a limit of the rescaling sequence \left(X, \frac{1}{r_n}d\right). The set of all pretangent spaces \Omega_{\infty, \tilde r}^{X} is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph (G_{X, \tilde r}, \rho_{X}) whose maximal cliques coincide with \Omega_{\infty, \tilde r}^{X} and the weight \rho_{X} is defined by metrics on \Omega_{\infty, \tilde r}^{X}. We describe the structure …