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]

References

↑ 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.
↑ 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.

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Quasi-relative interior

See also

References