Summary: A Minkowski-Rådström-Hörmander space \(\widetilde{X}\) is a quotient space over the family \(\mathcal{B}(X)\) of all non-empty bounded closed convex subsets of a Banach space \(X\). We prove in Theorem 4.2 that a metric \(d_{BP}\) (Bartels-Pallaschke metric) is the strongest of all complete metrics in the cone \(\mathcal{B}(X)\) and the Hausdorff metric \(d_H\) 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 \(S\) with complete metric \(d\). We also extend a definition of Demyanov’s difference (related to Clarke’s subdifferential) of finite-dimensional convex sets \(A \overset{D}{-} B\) to an infinite dimensional Banach space \(X\) and we prove in Theorem 4.1 that Demyanov’s metric generated by such extension, is complete.