is used broadly in developing generalized differentiation theory for convex extended-real-valued functions. For instance, the fundamental result of convex analysis, known as the Moreau-Rockafellar theorem, says that the subdifferential of the sum of two convex functions on $\R^n$ at a point where both functions are finite is represented as the Minkowski sum of the subdifferentials of each functions at references point [*provided*]{} that intersection of the relative interiors of the domains of these functions is nonempty. Another fundamental result of finite-dimensional geometry tells us the empty intersection of relative interiors of two convex sets is a [*characterization*]{} of [*proper separation*]{} of sets. In convex optimization and related topics, the notion of relative interior plays a crucial role in resolving many principal issues of the theory and applications including Fenchel duality, Lagrange duality, optimality conditions, numerical algorithms, etc.; see, e.g., [@Bauschke2011; @Bertsekas2003; @Borwein2000; @Boyd2004; @g; @HP; @HU; @KK2013; @bmn; @pr; @rw] and the references therein.

It has been recognized for a long time that the classical notion of interior for convex sets and related results in infinite-dimensional spaces, based on convex separation under nonempty interior assumptions, create–besides being rather restrictive in theoretical developments–serious limitations for applications. Among various areas that suffer 