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 $$\operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}$$ where $\operatorname {\overline {cone}} (\cdot )$ 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 $\operatorname {qri} (C)=\operatorname {ri} (C)$ such that $\operatorname {ri}$ 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.