Quasi-relative interior

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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 algebraic interior of A \subseteq X is

\operatorname{qri}(A) := \left\{x \in A: \overline{\operatorname{cone}}(A - x) \text{ is a linear subspace}\right\} \,

where \overline{\operatorname{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]

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References[edit]

  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.