We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov– Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our i...

We introduce Lipschitz continuity of set-valued maps with respect to a given set difference. The existence of Lipschitz selections that pass through any point of the graph of the map and inherit its Lipschitz constant is studied. We show that the Lipschitz property of the set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of its ge...

If X is a complete metric space, the collection of all non-empty compact subsets of X forms a complete metric space (H(X), h), where h is the Hausdorff metric. In this paper we explore some of the geometry of the space H(Rn). Specifically, we concentrate on understanding lines in H(R). In particular, we show that for any two points A, B ∈ H(Rn), there exist infinitely many points on the line jo...

Abstract We show continuity under equivariant Gromov–Hausdorff convergence of the critical exponent discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on Gromov-hyperbolic metric spaces.

Given a compact and connected metric space (continuum) X, we study topological and dynamical properties of the hyperspace of normal fuzzy sets F1(X) equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph met...

This paper investigates superspaces 𝒫0(X) and 𝒦0(X) of a tvs-cone metric space (X, d), where 𝒫0(X) and 𝒦0(X) are the space consisting of nonempty subsets of X and the space consisting of nonempty compact subsets of X, respectively. The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology (resp., the upper topology and the...

We present geometric conditions on a metric space (Y, dY ) ensuring that almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov-Hausdorff limits, and Cartesian products. ...

We characterize the convergence of fuzzy sets in the supremum metric given by the supremum of the Hausdorff distances of the α-cuts of the fuzzy sets. We do it by dividing this metric into its lower and upper quasipseudometric parts. This characterization is given in the more general context with no assumption on the fuzzy sets. Furthermore, motivated from the theory of Convex Analysis, we also...