  This paper focuses on investigating generalized relative interior notions for
sets in locally convex topological vector spaces with particular attentions to
graphs of set-valued mappings and epigraphs of extended-real-valued functions.
We introduce, study, and utilize a novel notion of quasi-near convexity of sets
that is an infinite-dimensional extension of the widely acknowledged notion of
near convexity. Quasi-near convexity is associated with the quasi-relative
interior of sets, which is investigated in the paper together with other
generalized relative interior notions for sets, not necessarily convex. In this
way, we obtain new results on generalized relative interiors for graphs of
set-valued mappings in convexity and generalized convexity settings.
