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Algebraic interior

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Title: Algebraic interior  
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Algebraic interior

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 that it is absorbing with respect to, 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 X is a linear space then the algebraic interior of A \subseteq X is

\operatorname{core}(A) := \left\{x_0 \in A: \forall x \in X, \exists t_x > 0, \forall t \in [0,t_x], x_0 + tx \in A\right\}.[4]

Note that in general \operatorname{core}(A) \neq \operatorname{core}(\operatorname{core}(A)), but if A is a convex set then \operatorname{core}(A) = \operatorname{core}(\operatorname{core}(A)). In fact, if A is a convex set then if x_0 \in \operatorname{core}(A), y \in A, 0 < \lambda \leq 1 then \lambda x_0 + (1 - \lambda)y \in \operatorname{core}(A).

Contents

  • Example 1
  • Properties 2
    • Relation to interior 2.1
  • See also 3
  • References 4

Example

If A \subset \mathbb{R}^2 such that A = \{x \in \mathbb{R}^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\} then 0 \in \operatorname{core}(A), but 0 \not\in \operatorname{int}(A) and 0 \not\in \operatorname{core}(\operatorname{core}(A)).

Properties

Let A,B \subset X then:

  • A is absorbing if and only if 0 \in \operatorname{core}(A).[1]
  • A + \operatorname{core}B \subset \operatorname{core}(A + B)[5]
  • A + \operatorname{core}B = \operatorname{core}(A + B) if B = \operatorname{core}B[5]

Relation to interior

Let X be a topological vector space, \operatorname{int} denote the interior operator, and A \subset X then:

  • \operatorname{int}A \subseteq \operatorname{core}A
  • If A is nonempty convex and X is finite-dimensional, then \operatorname{int}A = \operatorname{core}A[2]
  • If A is convex with non-empty interior, then \operatorname{int}A = \operatorname{core}A[6]
  • If A is a closed convex set and X is a complete metric space, then \operatorname{int}A = \operatorname{core}A[7]

See also

References

  1. ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and (\mu,\rho)-Portfolio Optimization". 
  2. ^ a b Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. pp. 199–200.  
  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.  
  5. ^ a b Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ,: World Scientific Publishing  Co., Inc. pp. 2–3.  
  6. ^ Shmuel Kantorovitz (2003). Introduction to Modern Analysis. Oxford University Press. p. 134.  
  7. ^ Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56,  .
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