Quasi-relative interior

In topology, a branch of mathematics, the quasi-relative interior of a subset of a vector space is a refinement of the concept of the interior. Formally, if $X$ is a linear space then the quasi-relative interior of $A\subseteq X$ is $$\mathrm{qri}(A):=\{x\in A:\overline{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}}(A-x)\text{ is a linear subspace}\}$$ where $\overline{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}}(\cdot )$ denotes the closure of the conic hull. Let $X$ be a normed vector space. If $C\subseteq X$ is a convex finite-dimensional set then $\mathrm{qri}(C)=\mathrm{ri}(C)$ such that $\mathrm{ri}$ is the relative interior.

See also

• Interior (topology) – Largest open subset of some given set
• Relative interior – Generalization of topological interior
• Algebraic interior – Generalization of topological interior

References

• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.