it follows from that $\|u_k\|\to 0$, which clearly contradicts the condition $\|u_k\|\to 1$ as $k\to\infty$. This tells us that there exists $\ox\in\Omega$ such that $\la a,\ox\ra<0$, and hence $a\ne 0$. It justifies the proper separation of $\Omega$ and $\{0\}$ in and shows therefore that $0\notin\qri(\Omega)$.\
[**Step 2.**]{} To verify the quasi-regularity of $\Omega$, we need to prove that $\mbox{\rm qri}(\Omega)\subset\mbox{\rm iri}(\Omega)$. Picking any $\ox\in \mbox{\rm qri}(\Omega)$ gives us $0\in\qri(\Omega-\ox)$. Since $\Omega$ is SNC, this property holds for $\Omega-\ox$ as well. It allows us to deduce from the proof in Step 1 that $0\in\mbox{\rm iri}(\Omega-\ox)$, and so $\ox\in\mbox{\rm iri}(\Omega)$. This shows that $\mbox{\rm qri}(\Omega)\subset\mbox{\rm iri}(\Omega)$ and thus completes the proof of the theorem. $\h$

Quasi-Relative Interiors of Convex Graphs
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In this section we start studying properties of set-valued mappings with convex graphs that involve the (extended) relative interior constructions defined above. Remember that a set-valued mapping $F\colon X\tto Y$ between LCTV spaces is associated with its [*graph*]{} $$\gph(F):=\big\{(x,y)\in X\times Y\;\big|\;y\in F(x)\big\},$$ and it is called [*convex*]{} if its graph is a convex subset of the product space $X\times Y$. We also consider the [*domain*]{} of $F$ defined by $$\dom(F):=\big\{x\in X\;\big|\;F(x)\ne\emp\big\}.$$

Let us first present a new proof, based on proper 