X^*\times Y^*$ such that $$\label{equaRck2}
\la u^*,x\ra+\la v^*,y\ra\le\la u^*,\ox\ra+\la v^*,\oy\ra\;\mbox{ whenever }\;y\in F(x)$$ and also the existence of $(x_0,y_0)\in\gph(F)$ for which $$\la u^*,x_0\ra+\la v^*,y_0\ra<\la u^*,\ox\ra+\la v^*,\oy\ra.$$ Letting $x=\ox$ in yields $\la v^*,y\ra\le\la v^*,\oy\ra$ for all $y\in F(\ox)$. Using $\ox\in\qri\big(\dom(F)\big)$, $x_0\in\dom(F))$, and the assumed quasi-regularity of $\dom(F)$ allows us to deduce from Theorem \[Thrpriqri\](a) the existence of $\tilde{x}\in\dom(F)$ that ensures the representation $\ox=tx_0+(1-t)\tilde{x}$ with some $t\in(0, 1)$. Pick $\tilde{y}\in F(\tilde{x})$ and define $$y^\prime:=ty_0+(1-t)\tilde{y}.$$ Then $y^\prime\in F(\ox)$ by the convexity of $\gph(F)$, and we get $$\begin{aligned}
&\la u^*,\tilde{x}\ra+\la v^*,\tilde{y}\ra\le\la u^*,\ox\ra+\la v^*,\oy\ra,\\
&\la u^*,x_0\ra+\la v^*,y_0\ra<\la u^*,\ox\ra+\la v^*,\oy\ra.\end{aligned}$$ Multiply the first inequality above by $1-t$, the second inequality by $t$, and add them together to arrive at the condition $$\la u^*,\ox\ra+\la v^*,y^\prime\ra<\la u^*,\ox\ra+\la v^*,\oy\ra,$$ which gives us $\la v^*,y^\prime\ra<\la v^*,\oy\ra$. Thus we conclude that the sets $\{\oy\}$ and $F(\ox)$ are properly separated, and so $\oy\notin\qri(F(\ox))$ by Lemma \[proper separation\]. This contradiction shows that $(\ox,\oy)\in\qri(\gph(F))$ and hence completes the proof of the theorem. $\h$

Based on Theorem \[Lemalinearmap\], we arrive now at the precise calculation of the quasi-relative interiors of convex graphs for set-valued mappings $F\colon X\tto Y$ between LCTV spaces.

[**(precise calculation of quasi-relative interiors of convex graphs in LCTV spaces).**]{}\[qri-precise\] Let $F\colon X\tto 