proximaldomain

Proximaldomain is a term that appears in discussions of proximal mappings and proximinal sets within convex and variational analysis. While not universally standardized, it is used informally to denote the domain in which a proximal construction—such as a proximal projection or a proximal map with respect to a function or a set—can be carried out and yield a minimizing point.

Definition and interpretation. In a normed space X, consider a proper lower semicontinuous convex function f: X → (−∞, ∞]. The proximal map is defined by prox_f(x) = argmin_y { f(y) + (1/2) ||y − x||^2 }. For proper lsc convex f, prox_f is well-defined for all x in X, so the proximal domain in this sense would be all of X. When the focus is a subset A ⊂ X, the proximal projection onto A is defined by argmin_{y ∈ A} ||x − y||. If A is proximinal, such a minimizer exists for every x ∈ X, and the set of all x for which a proximal projection exists is called the proximal domain of A. In typical settings—such as A closed and convex in a Hilbert space—proximal projections exist for every x, making the proximal domain equal to X.

Relation and usage. The concept underlies many algorithms in convex optimization and variational analysis, where knowing the proximal domain helps assess feasibility and convergence. The proximal operator, Moreau envelope, and the theory of proximinal sets are closely related and provide the rigorous foundation behind these ideas.

Notes. The term proximaldomain is not universally adopted and may be encountered as an informal label in some texts. Standard references typically discuss proximal mappings, proximal projections, and proximinal sets directly.