thus $\qri(\Omega_1)\cap\qri(\Omega_2)=\emp$, which completes the proof. $\h$

Next we consider epigraphs of extended-real-valued convex functions and present far-going LCTV space extensions of the finite-dimensional result of Corollary \[epi-roc\].

[**(quasi-relative interiors of convex epigraphs in LCTV spaces).**]{}\[quasi epi\] Let $f\colon X\to\oR$ be a convex function, where $X$ is an LCTV space. Then we have $$\label{qriepi1}
{\rm qri}\big(\epi(f)\big)\supset\big\{(x,\lambda)\in X\times\R\;\big|\;x\in{\rm qri}\big(\dom(f)\big),\;\lambda>f(x)\big\}.$$ If we assume in addition that $\epi(f)$ is quasi-regular, then $$\label{qriepi}
{\rm qri}\big(\epi(f)\big)=\big\{(x,\lambda)\in X\times\R\;\big|\;x\in{\rm qri}\big(\dom(f)\big),\;\lambda>f(x)\big\}.$$

[**Proof.**]{} Note that follows from [@ZO Corollary 9(iii)]. Let $F\colon X\tto \R$ be defined by $F(x):=[f(x), \infty)$. Then $\gph(F)=\epi(f)$, $\dom(F)=\dom(f)$, and for $x\in \dom(f)$, we have $\qri(F(x))=(f(x), \infty)$. Under the quasi-regularity of $\gph(F)=\epi(f)$, the inclusion “$\subset$” in follows directly from Theorem \[rocthm2\](a). $\h$

\[affine reg\] Let $\Omega$ be a convex set in an LCTV space $X$. Given $x^*\in X^*$ and $b\in \R$, define the extended-real-valued function $$f(x):=\left\{\begin{aligned}
  &\la x^*,x\ra +b\; & \text{if} &\; x\in\Omega,\\
         &\infty \; &  \text{if}\;  &x\notin\Omega.
                \end{aligned}\right.$$ Then we have the representation $$\label{qrinotre}
  \qri(\epi(f))= \{(x,\lambda)\in X\times \mathbb{R}\ |\ x\in\qri(\Omega), \lambda>f(x)\}.$$

[**Proof.**]{} By in Theorem \[quasi epi\], it suffices to prove the inclusion “$\subset$” in . Taking $(x_0,\lambda_0)\in \qri(\epi(f)),$ we will show $(x_0,\lambda_0)$ belongs to the set on the right-hand side of . Define $F\colon 