Metric Spaces with Expensive Distances

Nearly equal distances in metric spaces

Let (X, d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9 logn n2 . This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound. We discus related questions and consider in part...

Banach-Like Distances and Metric Spaces of Compact Sets

In the first part we study general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space; we obtain a rigidity result. We then discuss the Hausdorff distance, proposing some less–known but important results: a closed–form formula for geodesics; generically two compact sets are connected by a continuum of geodesics. In the second p...

On metric spaces induced by fuzzy metric spaces

For a class of fuzzy metric spaces (in the sense of George and Veeramani) with an H-type t-norm,  we present a method to construct a metric on a  fuzzy metric space. The induced metric space shares many important properties with the given fuzzy metric space.  Specifically, they generate the same topology, and have the same completeness. Our results can give the constructive proofs to some probl...

Fixed Point Theorems for General Contractive Mappings with W-distances in Metric Spaces

In this paper, using the concept of w-distances on a metric space, we first prove a generalized fixed point theorem for mappings without continuity in a complete metric space. Using this result, we obtain new and well-known fixed point theorems in a complete metric space.

Existence theorems for single-valued and set-valued mappings with w-distances in metric spaces