relative interiors for such graphs in finite dimensions. The quasi-interior case of infinite-dimensional spaces is significantly more involved, the results obtained in this section are diverse and require imposing quasi-regularity assumptions. Further developments in this direction for convex sets, set-valued mappings, and extended-real-valued functions are given in Section 5.

Preliminaries
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Here we first recall the standard notation and definitions of convex analysis in LCTV spaces; see, e.g., [@z]. Then we present, for the reader’s convenience, some useful facts on the relative interior of convex sets in finite-dimensional spaces to compare them later with our new results via quasi-relative interiors in general infinite-dimensional settings.

Given a real LCTV space $X$ and its topological dual $X^*$, consider the canonical pairing $\la x^*,x\ra:=x^*(x)$ with $x\in X$ and $x^*\in X^*$. For a nonempty subset $A$ of $X$, define the [*conic hull*]{} of $A$ by $\cone(A):=\big\{ta\in X\;|\;t\ge 0,\;a\in A\big\}$ and denote the [*closure*]{} of $A$ by $\Bar{A}$.

Let $\Omega$ be a convex subset of $X$, and let $\ox\in\Omega$. The [*normal cone*]{} to $\Omega$ at $\ox$ is defined by $$\label{nc}
N(\ox;\Omega):=\big\{x^*\in X\;\big|\;\la x^*,x-\ox\ra\le 0\;\text{ for all }\;x\in\Omega\big\}$$ with $N(\ox;\Omega):=\emp$ if $\ox\notin\Omega$. Recall further that two convex sets $\Omega_1,\Omega_2\subset X$ are [properly separated]{} if there exists 