New types of convergence for unbounded star-shaped sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912458764976128 |
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| author | Higueras-Montaño, Luisa F. |
| author_facet | Higueras-Montaño, Luisa F. |
| contents | 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 $τ_{W^r}$ is not metrizable on $\mathcal{S}_{rc}^d$, while our radial Attouch-Wets type topology $τ_{AW^r}$ 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 $τ_{W^r}$ and $τ_{AW^r}$, and to establish topological properties of the family of flowers associated with closed convex sets containing the origin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New types of convergence for unbounded star-shaped sets Higueras-Montaño, Luisa F. Metric Geometry General Topology Geometric Topology 52A20, 52A30, 54A10, 54B20 (Primary) 40A30, 54C10 (Secondary) 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 $τ_{W^r}$ is not metrizable on $\mathcal{S}_{rc}^d$, while our radial Attouch-Wets type topology $τ_{AW^r}$ 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 $τ_{W^r}$ and $τ_{AW^r}$, and to establish topological properties of the family of flowers associated with closed convex sets containing the origin. |
| title | New types of convergence for unbounded star-shaped sets |
| topic | Metric Geometry General Topology Geometric Topology 52A20, 52A30, 54A10, 54B20 (Primary) 40A30, 54C10 (Secondary) |
| url | https://arxiv.org/abs/2507.00060 |