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 [math]\displaystyle{ X }[/math] is a linear space then the quasi-relative interior of [math]\displaystyle{ A \subseteq X }[/math] is [math]\displaystyle{ \operatorname{qri}(A) := \left\{x \in A : \operatorname{\overline{cone}}(A - x) \text{ is a linear subspace}\right\} }[/math] where [math]\displaystyle{ \operatorname{\overline{cone}}(\cdot) }[/math] denotes the closure of the conic hull.[1]
Let [math]\displaystyle{ X }[/math] is a normed vector space, if [math]\displaystyle{ C \subseteq X }[/math] is a convex finite-dimensional set then [math]\displaystyle{ \operatorname{qri}(C) = \operatorname{ri}(C) }[/math] such that [math]\displaystyle{ \operatorname{ri} }[/math] 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
- ↑ Zălinescu 2002, pp. 2-3.
- ↑ 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. http://legacy.orie.cornell.edu/~aslewis/publications/92-partially-I.pdf. Retrieved October 19, 2011.
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