标题： 无界星形集的新类型收敛 显示英文标题

标题： New types of convergence for unbounded star-shaped sets

摘要： 我们引入了Wijsman和Attouch-Wets拓扑的径向变体，用于星形集合$\mathcal{S}_{rc}^d$，这些集合$A \subseteq \mathbb{R}^d$是径向闭合的。这些拓扑为以原点为中心的星形集合提供了新的收敛类型，即使这些集合不是闭合或有界的。 我们的方法依赖于一类新的泛函，称为\textit{径向距离泛函}，它们测量点$x \in \mathbb{R}^d$与集合$A \in \mathcal{S}_{rc}^d$之间的“径向距离”。 这些是经典距离泛函的自然径向类比。 我们证明了我们的径向Wijsman型拓扑$\tau_{W^r}$在$\mathcal{S}_{rc}^d$上不是可度量化的，而我们的径向Attouch-Wets型拓扑$\tau_{AW^r}$是完全可度量化的。引入了一个相应的径向Attouch-Wets距离$d_{AW^r}$，并且我们证明了对于所有闭集$A,K \in \mathcal{S}_{rc}^d$，$d_{AW}(A,K) \leq d_{AW^r}(A,K)$成立，其中$d_{AW}$表示Attouch-Wets距离。 其中，这些结果被用来证明在$\mathcal{S}_{rc}^d$上关于$\tau_{W^r}$和$\tau_{AW^r}$的星对偶的连续性，并建立与包含原点的闭凸集相关的花族的拓扑性质。 显示英文摘要

摘要： 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.