from such limitations, we mention vector and set optimization as well as general equilibrium theory in economics; see [@bao-mor; @ktz; @mas; @m-book1; @m-book] for more discussions and references. In particular, it is well known that the ordering/positive cones in the classical Lebesgue spaces $l^p$ and $L^p$ for $1\le p<\infty$, which naturally appear in economic modeling, have empty interiors, and thus convex separation results conventionally used to establish appropriate price equilibria in basic economic models (e.g., in models of welfare economics) cannot be applied in such frameworks. Moreover, positive cones in the aforementioned and many other infinite-dimensional spaces important for applications have even [*empty relative interiors*]{}.

All of it provides a strong motivation to seek adequate counterparts of the relative interior notion for convex sets in infinite-dimensions. The major attention in this paper is paid to the notion of [*quasi-relative interior*]{} for convex sets that was introduced by Borwein and Lewis [@bl] and was studied and applied in various publications among which we particularly mention [@bao-mor; @BG; @bot; @daniele; @durea; @ktz; @m-book]. A key result of [@bl] tells us that the quasi-relative interior is [*nonempty*]{} for every nonempty closed and convex subset of any [*separable*]{} Banach space. Our main intention here 