Distance Functions Between Sets in (q1, q2)-Quasimetric Spaces

In this chapter we study the implications of our general metrization theory at the level of quasimetric spaces, with special emphasis on analytical aspects. More specifically, we study the nature of Holder functions on quasimetric spaces by proving density, embeddings, separation, and extension theorems. We also quantify the richness of such spaces by introducing and studying a notion of index that interfaces tightly with the critical exponent beyond which the Holder spaces become trivial. Other applications are targeted to Hardy spaces on spaces of homogeneous type, regularized distance, Whitney decompositions, and partitions of unity, as well as the Gromov–Pompeiu–Hausdorff distance.

On entropy on quasi-metric spaces

Kantorovich-Rubinstein quasi-metrics II: Hyperspaces and powerdomains

The ambition of this work is to introduce the notion of left (right) K-sequentially complete ordered dislocated fuzzy quasimetric spaces and to define a relevant Hausdorff metric on compact sets. A new approach, given in (Shoaib et al. in Filomat 34(2):323–338, 2020) has been used to obtain fixed-point results for multivalued mappings fulfilling generalized contraction in the latest framework. For the authenticity of our result, an example is formulated.

Hyperspaces of a weightable quasi-metric space: Application to models in the theory of computation

We study the Gromov–Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov–Hausdorff equivariant convergence and a version of Grove–Petersen–Wu's finiteness theorem for stratified spaces.

Quantized Gromov–Hausdorff distance

Gromov–Hausdorff convergence of non-Archimedean fuzzy metric spaces