Completeness in Minkowski–Rådström–Hörmander spaces

Abstract

A Minkowski–Rådström–Hörmander space is a quotient space over the family of all non-empty bounded closed convex subsets of a Banach space We prove in Theorem 4.2 that a metric (Bartels–Pallaschke metric) is the strongest of all complete metrics in the cone and Hausdorff metric is the coarsest of them. Our results follow from Theorem 3.1 for the more general case of a quotient space over an abstract convex cone with complete metric . We also extend a definition of Demyanov’s difference (related to Clarke’s subdifferential) of finite-dimensional convex sets to infinite dimensional Banach space and we prove in Theorem 4.1 that Demyanov’s metric generated by such extension, is complete.

Keywords:

• abstract convex cone
• translation invariant complete metric
• set difference
• Minkowski–Rådström–Hörmander space

AMS Subject Classifications:

• 52A07
• 22A20
• 46N10