Hyperspace convergences, bornologies and geometric set functionals

Key Points

Weak convergence depends on distance functionals, focusing on continuity and limits in metric space dynamics.
A relationship between uniform convergence and tau_{S,d}-convergence can illuminate underlying mathematical structures.
Investigation utilizes pointwise enlargement via a positive Lipschitz function to connect various convergence types.
Findings contribute new insights on Attouch-Wets convergence within established frameworks, enhancing theoretical understanding.

Abstract

For a bornology S of subsets of a metric space (X, d), we consider the following unified approaches of hyperspace convergence: convergence induced through uniform convergence of distance functionals (ₒ, ₃-convergence) ; bornological convergence, and the weak convergence induced by a family of gap and excess functionals. An interesting problem regarding these convergences is to investigate when any two of them are equivalent. In this article, we investigate the relation of ₒ, ₃-convergence with the other two convergences, which is not completely transparent. As a main tool for our investigation, we use the idea of pointwise enlargement of a set by a positive Lipschitz function. As applications of our results, we provide new proofs of some known results about Attouch-Wets convergence.

Hyperspace convergences, bornologies and geometric set functionals

Key Points

Abstract

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