point $x\in\Omega$ there exists another point $u\in\Omega$ such that $\ox\in(x,u)$.\
[(b)]{} $\ox\in\qri(\Omega)$ if and only if the normal cone $N(\ox;\Omega)$ is a subspace of $X^*$.

[**Proof.**]{} The equivalence in (a) follows from [@BG Lemma 2.3], while the one in (b) is proved in [@bl Proposition 2.8]. $\h$

Our next goal is to provide some sufficient conditions that ensure the quasi-regularity property of convex sets. To proceed, we recall a fundamental notion of general variational analysis in infinite-dimensional spaces that plays a crucial role in furnishing limiting procedures; see [@m-book1] for a comprehensive study and applications in convex and nonconvex settings.

[**(sequential normal compactness of sets).**]{}\[snc\] A nonempty closed convex subset $\Omega$ of a normed space $X$ is called [sequentially normally compact]{} ${\rm(SNC)}$ [at]{} $\ox\in\Omega$ if for any sequence $\{x_k\}$ in $\Omega$ converging to $\ox$ we have the implication $$\big[x^*_k\in N(x_k;\Omega),\;x^*_k\xrightarrow{w^*}0\big]\Longrightarrow\big[\|x^*_k\|\to 0\big]\;\mbox{ as }\;k\to\infty,$$ where symbol $w^*$ indicates the weak$^*$ topology on $X^*$. The set $\Omega$ is said to be [SNC]{} if it has this property at any point $\ox\in\Omega$.

A (convex) set $\Omega\subset X$ is surely SNC if $\mbox{\rm int}(\Omega)\ne\emp$; see [@m-book1 Proposition 1.25]. If $X$ is Banach and $\mbox{\rm ri}(\Omega)\ne\emp$, we have the following more delicate characterization: $\Omega$ is 