Abstract We introduce radial variants of the Wijsman and Attouch-Wets topologies for the family Sₑ₂^d S r c d of star-shaped sets (with respect to the origin) in R^d 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 Sₑ₂^d 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 ₖ^ₑ τ W r is not metrizable on Sₑ₂^d S r c d, while our radial Attouch-Wets type topology ₀ₖ^ₑ τ A W r is completely metrizable. A corresponding radial Attouch-Wets distance d₀ₖ^ₑ d A W r is introduced, and we prove that d₀ₖ (A, K) d₀ₖ^ₑ (A, K) d A W (A, K) ≤ d A W r (A, K) for all closed A, K Sₑ₂^d A, K ∈ S r c d, where d₀ₖ d A W denotes the Attouch-Wets distance.
Luisa F. Higueras-Montaño (Fri,) studied this question.