follows directly from the definitions that for any convex set $\Omega\subset X$ we have $$\ri(\Omega)\subset\iri(\Omega)\subset\qri(\Omega).$$ In Definition \[qri\](c) we designate the property $\qri(\Omega)={\rm iri}(\Omega)$ by labeling the sets satisfying this condition as [*quasi-regular*]{} ones. The latter property plays an important role in the subsequent results of the paper. Some sufficient conditions for the quasi-regularity property of convex sets are presented below.

The following example taken from [@BG] demonstrates that the sets $\qri(\Omega)$ and $\iri(\Omega)$ may be different in rather simple situations in the framework of Hilbert spaces.

[**(difference between quasi-relative and intrinsic relative interiors).**]{}\[ri-diff\] [Consider the classical space of sequences $X:=\ell^2$ and its convex subset $$\Omega:=\Big\{x=(x_k)\in X\;\Big|\;\|x\|_1:=\sum_{k=1}^\infty|x_k|\le 1\Big\}.$$ Then we have $\mbox{\rm iri}(\Omega)=\{x\in X\;|\;\|x\|_1<1\}$, while $$\qri(\Omega)=\Omega\setminus\big\{x=(x_k)\in X\;\big|\;\|x\|_1=1,\;x_k=0\;\mbox{ for all }\;k\ge k_0\;\mbox{\rm with some }\;k_0\in\N\big\}.$$]{}

The next theorem provides some characterizations of intrinsic relative and quasi-relative interiors of convex sets in LCTV spaces and shows that the equivalences (b)$\Longleftrightarrow$(c) and (d)$\Longleftrightarrow$(e) of Theorem \[ri\] hold true in infinite dimensions.

[**(characterizations of intrinsic relative and quasi-relative interiors in LCTV space).**]{}\[Thrpriqri\] Let $X$ be an LCTV space, and let $\Omega$ be a nonempty convex subset of $X$ with $\ox\in\Omega$. Then we have the equivalences:\
[(a)]{} $\ox\in\mbox{\rm iri}(\Omega)$ if and only if for every 