are considered in the next section.

Intrinsic Relative and Quasi-Relative Interiors
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We start with the following basic definitions used throughout the whole paper.

[**(extended relative interiors in infinite dimensions).**]{}\[qri\] Let $X$ be an LCTV space, and let $\Omega$ be a nonempty convex subset of $X$. Then:\
[(a)]{} The [intrinsic relative interior]{} of $\Omega$ is the set $$\mbox{\rm iri}(\Omega):=\big\{x\in\Omega\;\big|\;\mbox{\rm cone}(\Omega-x)\;\mbox{\rm is a subspace of }\;X\big\}.$$ [(b)]{} The [quasi-relative interior]{} of $\Omega$ is the set $$\mbox{\rm qri}(\Omega):=\big\{x\in\Omega\;\big|\;\Bar{\mbox{\rm cone}(\Omega-x)}\;\mbox{\rm is a subspace of }\;X\big\}.$$ [(c)]{} We say that a convex set $\Omega\subset X$ is [quasi-regular]{} if $\qri(\Omega)={\rm iri}(\Omega)$.

Due to Theorem \[pts2\], both notions in Definition \[qri\](a,b) reduce to the relative interior of $\Omega$ in finite-dimensional spaces. The one in (a) has been known under the name “intrinsic core" [@hol] (which may be confusing; see [@BG]) and also under the name “pseudo-relative interior" [@BG], which seems to be confusing as well since “pseudo" means “false". Following [@bao-mor] and [@m-book], we prefer to use the term [*intrinsic relative interior*]{} of $\Omega$ and denote this set by $\iri(\Omega)$.

As mentioned in Section 1, the notion of [*quasi-relative interior*]{} for convex sets was introduced in [@bl] and then was studied therein and other publications. It 