In this paper we derive a generalizing concept of $G$-norms, which we call $G$-sets, which is used to characterize minimizers of non-differentiable regularization functionals. Moreover, the concept is closely related to the defnition of slopes as published in a recent book by Ambrosio, Gigli, Savaré. A paradigm of regularization models fitting in this framework is robust bounded variation regularization. Two essential properties of this regularization technique are documented in the literature and it is shown that these properties can also be achieved with metric regularization techniques.
"Slope and $G$-set characterization of set-valued functions and applications to non-differentiable optimization problems." Commun. Math. Sci. 3 (4) 479 - 492, December 2005.