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 ${\displaystyle X}$ is 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

References

1. Zălinescu 2002, pp. 2–3.
2. Borwein, J.M.; Lewis, A.S. (1992). "Partially finite convex programming, Part I: Quasi relative interiors and duality theory" (pdf). Mathematical Programming. 57: 15–48. doi:10.1007/bf01581072. Retrieved October 19, 2011.

• 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.