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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