Characterization of regularity via variational stability of alternating projections sequences

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.