Algebraic interior

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In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior. It is the subset of points contained in a given set with respect to which it is absorbing, i.e. the radial points of the set.[1] The elements of the algebraic interior are often referred to as internal points.[2][3]

Formally, if is a linear space then the algebraic interior of is

[4]

Note that in general , but if is a convex set then . If is a convex set then if then .

Example[edit]

If such that then , but and .

Properties[edit]

Let then:

  • is absorbing if and only if .[1]
  • [5]
  • if [5]

Relation to interior[edit]

Let be a topological vector space, denote the interior operator, and then:

  • If is nonempty convex and is finite-dimensional, then [2]
  • If is convex with non-empty interior, then [6]
  • If is a closed convex set and is a complete metric space, then [7]

See also[edit]

References[edit]

  1. ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ()-Portfolio Optimization". 
  2. ^ a b Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. pp. 199–200. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0. 
  3. ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf). Retrieved November 14, 2012. 
  4. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6. 
  5. ^ a b 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. 
  6. ^ Shmuel Kantorovitz (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568. 
  7. ^ Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057 .