conditions:\
[(a)]{} $X$ is a finite-dimensional space.\
[(b)]{} $X$ is an LCTV space and $\mbox{\rm int}(\Omega)\ne\emp$.\
[(c)]{} $X$ is an LCTV space and $\mbox{\rm ri}(\Omega)\ne\emp$.\
[(d)]{} $X$ is a Hilbert space, $\Omega$ is closed and SNC with $\iri(\Omega)\ne\emp$.

[**Proof.**]{} The quasi-regularity of $\Omega$ in case (a) is proved in Theorem \[pts2\]. The fulfillment of this property under (b) and (c) follows from [@BG Theorem 2.12]. Let us now show that $\Omega$ is quasi-regular under the assumptions in (d). We split the proof into the following two steps.

[**Step 1.**]{} First we verify that in the case where $0\notin{\rm iri}(\Omega)$, the sets $\Omega$ and $\{0\}$ are properly separated, i.e., there is a nonzero vector $a\in X$ such that $$\label{pro-sep}
\sup\big\{\langle a,x\rangle\;\big|\;x\in\Omega\big\}\le 0\;\mbox{ and }\;\inf\big\{\langle a,x\rangle\;\big|\;x\in\Omega\big\}<0,$$ which is equivalent therefore to $0\notin\qri(\Omega)$ by Lemma \[proper separation\].

If $0\not\in\Omega$, this statement is trivial. Suppose now that $0\in\Omega\setminus\text{\rm iri}(\Omega)$. Letting $L:=\overline{\mbox{\rm aff}(\Omega)}$ and employing Lemma \[lm2separate\] tell us that $L$ is a subspace of $X$ and that there exists a sequence $\{x_k\}\subset L$ for which $x_k\notin {\Omega}$ and $x_k\to 0$ as $k\to\infty$. By Lemma \[Proseparate1\] we find a sequence $\{v_k\}\subset L$ with $v_k\ne 0$ such that $$\sup\big\{\langle v_k,x\rangle\;\big|\;x\in\Omega\big\}<\la v_k,x_k\ra\;\mbox{ whenever }\;k\in\N.$$ Denote $u_k:=\frac{v_k}{\Vert v_k\Vert}\in 