New types of convergence for unbounded star-shaped 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 ${\tau }_{A{W}^r}$ is completely metrizable. A corresponding radial Attouch-Wets distance $d_{A{W}^r}$ is introduced, and we prove that $d_{AW}(A,K)\le d_{A{W}^r}(A,K)$ 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 ${\tau }_{A{W}^r}$, and to establish topological properties of the family of flowers associated with closed convex sets containing the origin.

35 pages