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 {\displaystyle X} is a linear space then the quasi-relative interior of A ⊆ X {\displaystyle A\subseteq X} is
qri ( A ) := { x ∈ A : c o n e ¯ ( A − x ) is a linear subspace } {\displaystyle \operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}}
where c o n e ¯ ( ⋅ ) {\displaystyle \operatorname {\overline {cone}} (\cdot )} denotes the closure of the conic hull. Let X {\displaystyle X} be a normed vector space. If C ⊆ X {\displaystyle C\subseteq X} is a convex finite-dimensional set then qri ( C ) = ri ( C ) {\displaystyle \operatorname {qri} (C)=\operatorname {ri} (C)} such that ri {\displaystyle \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.