About -regularity properties of collections of sets

We examine three primal space local Hِlder type regularity properties of finite collections of sets, namely, [ q ] -semiregularity, [ q ] -subregularity, and uniform [ q ] -re-gularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [ q ] -subregularity in terms of Fréchet normals are established. The relationships between [ q ] -regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.