The quasi‑relative interior (often abbreviated qri) is a refinement of the relative interior used in convex analysis, particularly in infinite‑dimensional vector spaces where the ordinary relative interior may be empty. It provides a notion of “interior points” relative to the affine hull of a set that is better behaved under closure operations.
Definition
Let $X$ be a real topological vector space and let $C\subseteq X$ be a non‑empty convex set.
The quasi‑relative interior of $C$ is
$$ \operatorname{qri}(C)=\Bigl{x\in C ;\Big|; \forall, y\in X,; \exists, \varepsilon>0 \text{ such that } x+\lambda y\in C\ \forall \lambda\in[0,\varepsilon]\Bigr}. $$
Equivalently, $x\in\operatorname{qri}(C)$ iff every direction $y$ that can be entered from $x$ while staying in the closure of $C$ actually belongs to the closure of the affine hull of $C$. Formally,
$$ \operatorname{qri}(C)=\Bigl{x\in C ;\Big|; \operatorname{cone}(C-x)\text{ is a linear subspace of }\operatorname{aff}(C)\Bigr}, $$
where $\operatorname{cone}(C-x)={\lambda (z-x)\mid z\in C,;\lambda\ge 0}$.
Relationship to Other Interior Notions
| Concept | Definition | Inclusion Relations (in general) |
|---|---|---|
| Interior $\operatorname{int}(C)$ | Points having a neighborhood contained in $C$. | $\operatorname{int}(C)\subseteq\operatorname{ri}(C)\subseteq\operatorname{qri}(C)$. |
| Relative interior $\operatorname{ri}(C)$ | Interior relative to $\operatorname{aff}(C)$. | $\operatorname{ri}(C)\subseteq\operatorname{qri}(C)$. |
| Quasi‑relative interior $\operatorname{qri}(C)$ | As defined above. | May be non‑empty even when $\operatorname{ri}(C)=\varnothing$. |
In finite‑dimensional spaces the three notions coincide:
$$ \operatorname{int}(C)=\operatorname{ri}(C)=\operatorname{qri}(C). $$
In infinite‑dimensional spaces, however, the quasi‑relative interior can be strictly larger than the relative interior and is often non‑empty for convex sets that have empty relative interior.
Main Properties
- Convexity Preservation: $\operatorname{qri}(C)$ is a convex subset of $C$.
- Closure Relation: $\operatorname{cl}(\operatorname{qri}(C)) = \operatorname{cl}(C)$ for closed convex $C$.
- Stability under Linear Transformations: If $A:X\to Y$ is a continuous linear map, then
$$ A(\operatorname{qri}(C))\subseteq \operatorname{qri}(A(C)). $$
Equality holds when $A$ is surjective. - Dual Characterization: For a convex set $C$,
$$ x\in\operatorname{qri}(C) \iff \forall, f\in X^{*}\ \bigl(f(x)=\inf_{z\in C}f(z)\implies f\text{ is constant on }C\bigr). $$ - Utility in Constraint Qualifications: In convex optimization, replacing the Slater condition (existence of a point in the interior) with a quasi‑relative interior condition yields constraint qualifications that are applicable in infinite‑dimensional problems.
Applications
- Convex Optimization – The quasi‑relative interior appears in generalized Slater‑type conditions, duality theorems, and the formulation of optimality conditions for problems in Banach spaces.
- Variational Analysis – It is used to describe regularity properties of set‑valued mappings and to formulate subdifferential calculus rules when standard interior assumptions fail.
- Functional Analysis – In the study of convex cones, especially those that are not closed, the qri provides a tractable “core” of the cone for separation theorems.
References
- Borwein, J. M., & Lewis, A. S. (2000). Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer. – Section on interior notions.
- Rockafellar, R. T. (1997). Convex Analysis. Princeton University Press. – Discusses relative interior; the quasi‑relative interior is treated in later extensions.
- Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. World Scientific. – Provides a systematic treatment of qri.
- Bot, C., Csetnek, G., & Zălinescu, C. (2010). “On the quasi‑relative interior of convex sets.” Set-Valued and Variational Analysis, 18(2), 263‑279.
These sources establish the quasi‑relative interior as a standard concept in modern convex analysis.
