interiors for epigraphs of extended-real-valued convex functions on $\R^n$. Recall that the [*domain*]{} and [*epigraph*]{} of a function $f\colon X\to\oR:=(-\infty,\infty]$ are defined, respectively, by $$\dom(f):=\big\{x\in X\;\big|\;f(x)<\infty\big\}\;\mbox{ and }\;\epi(f):=\big\{(x,\al)\in X\times\R\;\big|\;\al\ge f(x.)\big\}$$ The convexity of $f$ corresponds to the convexity of $\epi(f)$, and $f$ is [*proper*]{} if $\dom(f)\ne\emp$.

[**(relative interiors of epigraphs for convex functions on $\R^n$).**]{}\[epi-roc\] Let $f\colon\R^n\to\oR$ be a convex function. Then we have $$\label{epi}
\ri\big(\epi(f)\big)=\big\{(x,\lambda)\in\R^{n+1}\;\big|\;x\in\ri(\dom(f)),\;\lambda>f(x)\big\}.$$

[**Proof.**]{} Given $f\colon\R^n\to\oR$, define the set-valued mapping $F\colon\R^n\tto\R$ by $F(x):=\big[f(x),\infty\big)$ for all $x\in\R^n$. Then $\dom(F)=\dom(f)$, $\gph(F)=\epi(f)$, and for any $x\in\dom(f)$ we have $\ri(F(x))=\big(f(x),\infty\big)$. Then follows directly from Theorem \[TheoRoc1\]. $\h$

Next we extend Rockafellar’s theorem to the general setting of LCTV spaces. As the reader can see, the extension below, which formulated in terms of the quasi-relative interior, is more involved in infinite dimensions while giving us two inclusion counterparts of under different quasi-regularity assumptions. Our proof is based on the proper separation description of the quasi-relative interior established in Lemma \[proper separation\].

[**(quasi-relative interiors of convex graphs in LCTV space).**]{}\[rocthm2\] Let $F\colon X\tto Y$ be a convex set-valued mapping between LCTV spaces. The following hold:\
[(a)]{} If the graph $\gph(F)$ is quasi-regular, then we have the inclusion $$\mbox{\rm qri}\big(\gph(F)\big)\subset\big\{(x,y)\in X\times Y\;\big|\;x\in\mbox{\rm qri}\big(\dom(F)\big),\;y\in\mbox{\rm qri}\big(F(x)\big)\big\}.$$ 