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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.

Definition

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Assume that is a subset of a vector space The algebraic interior (or radial kernel) of with respect to is the set of all points at which is a radial set. A point is called an internal point of [1][2] and is said to be radial at if for every there exists a real number such that for every This last condition can also be written as where the set is the line segment (or closed interval) starting at and ending at this line segment is a subset of which is the ray emanating from in the direction of (that is, parallel to/a translation of ). Thus geometrically, an interior point of a subset is a point with the property that in every possible direction (vector) contains some (non-degenerate) line segment starting at and heading in that direction (i.e. a subset of the ray ). The algebraic interior of (with respect to ) is the set of all such points. That is to say, it is the subset of points contained in a given set with respect to which it is radial points of the set.[3]

If is a linear subspace of and then this definition can be generalized to the algebraic interior of with respect to is:[4] where always holds and if then where is the affine hull of (which is equal to ).

Algebraic closure

A point is said to be linearly accessible from a subset if there exists some such that the line segment is contained in [5] The algebraic closure of with respect to , denoted by consists of and all points in that are linearly accessible from [5]

Algebraic Interior (Core)

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In the special case where the set is called the algebraic interior or core of and it is denoted by or Formally, if is a vector space then the algebraic interior of is[6]

If is non-empty, then these additional subsets are also useful for the statements of many theorems in convex functional analysis (such as the Ursescu theorem):

If is a Fréchet space, is convex, and is closed in then but in general it is possible to have while is not empty.

Examples

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If then but and

Properties of core

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Suppose

  • In general, But if is a convex set then:
    • and
    • for all then
  • is an absorbing subset of a real vector space if and only if [3]
  • [7]
  • if [7]

Both the core and the algebraic closure of a convex set are again convex.[5] If is convex, and then the line segment is contained in [5]

Relation to topological interior

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

  • If is nonempty convex and is finite-dimensional, then [1]
  • If is convex with non-empty interior, then [8]
  • If is a closed convex set and is a complete metric space, then [9]

Relative algebraic interior

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If then the set is denoted by and it is called the relative algebraic interior of [7] This name stems from the fact that if and only if and (where if and only if ).

Relative interior

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If is a subset of a topological vector space then the relative interior of is the set That is, it is the topological interior of A in which is the smallest affine linear subspace of containing The following set is also useful:

Quasi relative interior

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If is a subset of a topological vector space then the quasi relative interior of is the set

In a Hausdorff finite dimensional topological vector space,

See also

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References

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  1. ^ a b Aliprantis & Border 2006, pp. 199–200.
  2. ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.
  3. ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ()-Portfolio Optimization" (PDF).
  4. ^ Zălinescu 2002, p. 2.
  5. ^ a b c d Narici & Beckenstein 2011, p. 109.
  6. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
  7. ^ a b c Zălinescu 2002, pp. 2–3.
  8. ^ Kantorovitz, Shmuel (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.
  9. ^ Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057.

Bibliography

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