is to study further properties of quasi-relative interior in [*locally convex topological vector*]{} (LCTV) spaces. We obtain a quasi-relative extension in LCVT spaces of Rockafellar’s finite-dimensional theorem on representing the relative interior of convex graphs for set-valued mappings via the relative interiors of the corresponding domain and image sets. The obtained quasi-relative version of such a representation in infinite-dimensional spaces requires a certain [*quasi-regularity*]{} condition introduced below, which automatically holds in finite dimensions. We also obtain new results on extended relative interiors of convex sets and epigraphs of convex functions on LCTV spaces.

The rest of the paper is organized as follows. Section 2 presents some preliminary material used in the sequel. Section 3 contains the definitions of and discussions on the basic notions of intrinsic relative interior, quasi-relative interior, and quasi-regularity for convex sets in LCTV spaces. Besides other results, we establish here relationships between these properties and the sequential normal compactness of convex sets, which is one the central notions of general infinite-dimensional variational analysis. Section 4 is devoted to the study of quasi-relative interiors for convex graphs of set-valued mappings between LCTV spaces. First we give an alternative proof of Rockafellar’s theorem concerning a precise representation of 