New types of convergence for unbounded star-shaped sets

TL;DR

This paper introduces new radial topologies for star-shaped sets in Euclidean space, establishing their properties and applications, including continuity of star duality and topological analysis of associated flower families.

Contribution

It develops radial variants of classical topologies for star-shaped sets, introduces radial distance functionals, and analyzes their metrizability and continuity properties.

Findings

Radial Wijsman topology is not metrizable.

Radial Attouch-Wets topology is completely metrizable.

Radial distances provide natural measures for convergence of star sets.

Abstract

We introduce radial variants of the Wijsman and Attouch-Wets topologies for the family $\mathcal{S}_{rc}^d$ of star sets $A \subseteq \mathbb{R}^d$ that are radially closed.These topologies give rise to new types of convergence for star-shaped sets with respect to the origin, even when such sets are not closed or bounded. Our approach relies on a new family of functionals, called \textit{radial distance functionals}, which measure ``radial distances'' between points $x \in \mathbb{R}^d$ and sets $A \in \mathcal{S}_{rc}^d$. These are natural radial analogues of the classical distance functionals. We prove that our radial Wijsman type topology $\tau_{W^r}$ is not metrizable on $\mathcal{S}_{rc}^d$, while our radial Attouch-Wets type topology $\tau_{AW^r}$ is completely metrizable. A corresponding radial Attouch-Wets distance $d_{AW^r}$ is introduced, and we prove that $d_{AW}(A,K) \leq d_{AW^r}(A,K)$ for all closed $A,K \in \mathcal{S}_{rc}^d$, where $d_{AW}$ denotes the Attouch-Wets distance. Among others, these results are applied to prove the continuity of the star duality on $\mathcal{S}_{rc}^d$ with respect to both $\tau_{W^r}$ and $\tau_{AW^r}$, and to establish topological properties of the family of flowers associated with closed convex sets containing the origin.