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.^[1]

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

See also

• Interior (topology)
• Relative interior
• Algebraic interior

References

1. Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing  Co., Inc. pp. 2–3. ISBN 981-238-067-1. MR 1921556.
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.