[**Quasi-Relative Interiors for Graphs of Convex Set-Valued Mappings**]{}\
[Dang Van Cuong]{}[^1], [Boris S. Mordukhovich]{}[^2], [Nguyen Mau Nam]{}[^3]

This paper aims at providing further studies of the notion of quasi-relative interior for convex sets introduced by Borwein and Lewis. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.\
[**Key words.**]{} quasi-relative interior, intrinsic relative interior, quasi-regularity, proper separation\
[**AMS subject classifications.**]{} 49J52, 49J53, 90C31

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Introduction
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The notion of [*relative interior*]{} for convex sets in finite-dimensional spaces was largely developed by Rockafellar in his seminal monograph “Convex Analysis" [@r] as a refinement of the classical notion of interior. In contrast to the latter, the relative interior is [*nonempty*]{} for any nonempty convex subset of $\R^n$. It has been fully recognized by now that results involving relative interior are highly important in many aspects of convex analysis and optimization in finite dimensions. In particular, relative interior 