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 ${\displaystyle X}$ is a linear space then the quasi-relative interior of ${\displaystyle A\subseteq X}$ is $${\displaystyle \operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}}$$ where ${\displaystyle \operatorname {\overline {cone}} (\cdot )}$ denotes the closure of the conic hull.^[1]

Let ${\displaystyle X}$ be a normed vector space. If ${\displaystyle C\subseteq X}$ is a convex finite-dimensional set then ${\displaystyle \operatorname {qri} (C)=\operatorname {ri} (C)}$ such that ${\displaystyle \operatorname {ri} }$ is the relative interior.^[2]

See also

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