New Types of Convergence for Unbounded Star-Shaped Sets

We introduce radial variants of the Wijsman and Attouch-Wets topologies for the family Srcᵈ S r c d of star-shaped sets (with respect to the origin) in Rᵈ R d that are radially closed. These topologies give rise to new types of convergence for star-shaped sets, even when such sets are not closed or bounded. Our approach relies on a new family of functionals, called radial distance functionals , which measure “radial distances” between points and star-shaped sets in Srcᵈ S r c d . These are natural radial analogues of the distance functionals for closed sets. However, unlike the radial functions of star-shaped sets, our radial distance functionals are real-valued maps and thus admit a natural treatment within the framework of classical function spaces. We prove that our radial Wijsman type topology τ _Wʳ τ W r is not metrizable on Srcᵈ S r c d , while our radial Attouch-Wets type topology τ _AWʳ τ A W r is completely metrizable. A corresponding radial Attouch-Wets distance d_AWʳ d A W r is introduced, and we prove that dAW(A,K) ≤ d_AWʳ(A,K) d A W ( A , K ) ≤ d A W r ( A , K ) for all closed A,K ∈ Srcᵈ A , K ∈ S r c d , where dAW d A W denotes the Attouch-Wets distance.