SNC if and only if the closure of its affine hull $\Bar{\aff}(\Omega)$ is [*finite-codimensional*]{}; see [@m-book1 Theorem 1.21]. Note that, in the framework of arbitrary normed spaces, a similar characterization is available for a generally more restrictive notion of the [*compactly epi-Lipschitzian*]{} (CEL) property for closed convex sets $\Omega$ with $\mbox{\rm ri}(\Omega)\ne\emp$: $\Omega$ is CEL if and only if its affine hull is a closed finite-codimensional subspace of $X$; see [@blm] and more discussions in [@m-book1 Remark 1.27].

Prior to deriving sufficient conditions for quasi-regularity, we present three lemmas of their own interest that are used in what follows. The first one is an intrinsic relative interior version in LCTV spaces of the well-known property for relative interiors in finite dimensions.

[**(convex sets with nonempty intrinsic relative interiors in LCTV spaces).**]{}\[lm2separate\] Let $X$ be an LCTV space, and let $\Omega\subset X$ be a nonempty closed convex set with $0\in\Omega\setminus\mbox{\rm iri}(\Omega)$. If ${\rm iri}(\Omega)\ne\emp$, then $\overline{\aff(\Omega)}$ is a closed subspace of $X$ and there is a sequence $\{x_k\}\subset -\Omega$ with $x_k\notin\Omega$ and $x_k\to 0$ as $k\to\infty$.

[**Proof.**]{} Using ${\rm iri}(\Omega)\ne\emp$ and $0\in\Omega\setminus\mbox{\rm iri}(\Omega)$, let us show that there exists a nonzero vector $x_0\in{\rm iri}(\Omega)$ such that $-tx_0\notin\Omega$ for all $t>0$. Arguing 