Y$ be a convex set-valued mapping between LCTV spaces. If both $\gph(F)$ and $\dom(F)$ are quasi-regular, then we have $$\mbox{\rm qri}\big(\gph(F)\big)=\big\{(x,y)\in X\times Y\;\big|\;x\in\mbox{\rm qri}\big(\dom(F)\big)\;y\in\mbox{\rm qri}\big(F(x)\big)\big\}.$$

Further Properties of Extended Relative Interiors
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In this section we continue our study of important properties of the quasi-relative and intrinsic relative interiors of convex set in LCTV spaces, with a particular emphasis on graphical, domain, and epigraphical sets for convex set-valued mappings and extended-real-valued functions. Note that the assumption $\qri(\Omega)\ne\emp$, which is imposed in many results presented in this section, automatically holds for closed convex sets in any separable Banach spaces due to the fundamental result of [@bl Theorem 2.19].

The first statement describes behavior of quasi-relative interiors of convex sets under applying linear continuous operators.

[**(quasi-relative interiors under linear mappings).**]{}\[Lemalinearmap\] Let $A\colon X\to Y$ be a linear continuous mapping between two LCTV spaces, and let $\Omega\subset X$ be a convex such that $\iri(\Omega)\neq\emptyset$ and $A(\Omega)$ is quasi-regular. Then we have $$A\big(\qri(\Omega)\big)=\qri\big(A(\Omega)\big).$$

[**Proof.**]{} The inclusion $A(\qri(\Omega))\subset\qri(A(\Omega))$ is known; see [@bl Proposition 2.21]. To verify the opposite inclusion, we use [@ktz Proposition 6.3.2(i)] and get $$\qri(A(\Omega))=\iri(A(\Omega))=A(\iri(\Omega))\subset A(\qri(\Omega)),$$ which completes the proof. $\h$

The next corollary on representing quasi-relative interiors of set differences is a 