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Hyperspace convergences, bornologies and geometric set functionals

Yogesh Agarwal, Varun Jindal

TL;DR

This paper examines how three unified approaches to hyperspace convergence for a bornology on a metric space relate to one another, focusing on tau_S,d-convergence, bornological convergence S, and gap-excess weak convergence. By leveraging the concept of pointwise Lipschitz enlargements, the authors derive conditions under which tau_S,d agrees with S-convergence and with gap-excess convergence, and they provide new proofs of Attouch-Wets convergence results as corollaries. Key contributions include necessary and sufficient conditions for tau_S,d to coincide with S and to align with LE and GE convergences under stability assumptions, along with a covering criterion that guarantees tau_S,d = tau_GE^S. These results deepen the understanding of the structural and topological relationships among hyperspace convergences and offer practical tools for variational analysis and related optimization contexts.

Abstract

For a bornology $\mathcal{S}$ 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 ($τ_{\mathcal{S},d}$-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 $τ_{\mathcal{S},d}$-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

TL;DR

This paper examines how three unified approaches to hyperspace convergence for a bornology on a metric space relate to one another, focusing on tau_S,d-convergence, bornological convergence S, and gap-excess weak convergence. By leveraging the concept of pointwise Lipschitz enlargements, the authors derive conditions under which tau_S,d agrees with S-convergence and with gap-excess convergence, and they provide new proofs of Attouch-Wets convergence results as corollaries. Key contributions include necessary and sufficient conditions for tau_S,d to coincide with S and to align with LE and GE convergences under stability assumptions, along with a covering criterion that guarantees tau_S,d = tau_GE^S. These results deepen the understanding of the structural and topological relationships among hyperspace convergences and offer practical tools for variational analysis and related optimization contexts.

Abstract

For a bornology of subsets of a metric space , we consider the following unified approaches of hyperspace convergence: convergence induced through uniform convergence of distance functionals (-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 -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.

Paper Structure

This paper contains 4 sections, 30 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. preliminaries
  3. $\tau_{\mathcal{S},d}$-convergence vis-á-vis $\mathcal{S}$-convergence
  4. $\tau_{\mathcal{S},d}$-convergence vis-á-vis gap and excess convergence

Key Result

Theorem 3.1

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$. Suppose $(A_{\lambda})$ is a net in $CL(X)$ and $A \in CL(X)$. Then the following statements are equivalent:

Figures (1)

  • Figure 1:

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1: Hit-type characterization for $\mathcal{S}^-$-convergence
  • proof
  • Theorem 3.2: Hit-and-Miss type characterization of $\mathcal{S}$-convergence
  • Proposition 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Example 3.6
  • ...and 56 more