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Relative interior

In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces.

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In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces.

Formally, the relative interior of a set S {\displaystyle S} (denoted relint ⁡ ( S ) {\displaystyle \operatorname {relint} (S)} ) is defined as its interior within the affine hull of S . {\displaystyle S.} 1 In other words, relint ⁡ ( S ) := { x ∈ S :  there exists  ϵ > 0  such that  B ϵ ( x ) ∩ aff ⁡ ( S ) ⊆ S } , {\displaystyle \operatorname {relint} (S):=\{x\in S:{\text{ there exists }}\epsilon >0{\text{ such that }}B_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},} where aff ⁡ ( S ) {\displaystyle \operatorname {aff} (S)} is the affine hull of S , {\displaystyle S,} and B ϵ ( x ) {\displaystyle B_{\epsilon }(x)} is a ball of radius ϵ {\displaystyle \epsilon } centered on x {\displaystyle x} . Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.

A set is relatively open iff it is equal to its relative interior. Note that when aff ⁡ ( S ) {\displaystyle \operatorname {aff} (S)} is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed.

For any convex set C ⊆ R n {\displaystyle C\subseteq \mathbb {R} ^{n}} the relative interior is equivalently defined as23 relint ⁡ ( C ) := { x ∈ C :  for all  y ∈ C ,  there exists some  λ > 1  such that  λ x + ( 1 − λ ) y ∈ C } = { x ∈ C :  for all  y ≠ x ∈ C ,  there exists some  z ∈ C  such that  x ∈ ( y , z ) } . {\displaystyle {\begin{aligned}\operatorname {relint} (C)&:=\{x\in C:{\text{ for all }}y\in C,{\text{ there exists some }}\lambda >1{\text{ such that }}\lambda x+(1-\lambda )y\in C\}\\&=\{x\in C:{\text{ for all }}y\neq x\in C,{\text{ there exists some }}z\in C{\text{ such that }}x\in (y,z)\}.\end{aligned}}} where x ∈ ( y , z ) {\displaystyle x\in (y,z)} means that there exists some 0 < λ < 1 {\displaystyle 0<\lambda <1} such that x = λ z + ( 1 − λ ) y {\displaystyle x=\lambda z+(1-\lambda )y} .

Comparison to interior

  • The interior of a point in an at least one-dimensional ambient space is empty, but its relative interior is the point itself.
  • The interior of a line segment in an at least two-dimensional ambient space is empty, but its relative interior is the line segment without its endpoints.
  • The interior of a disc in an at least three-dimensional ambient space is empty, but its relative interior is the same disc without its circular edge.

Properties

Theorem—If A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} is nonempty and convex, then its relative interior r e l i n t ( A ) {\displaystyle \mathrm {relint} (A)} is the union of a nested sequence of nonempty compact convex subsets K 1 ⊂ K 2 ⊂ K 3 ⊂ ⋯ ⊂ r e l i n t ( A ) {\displaystyle K_{1}\subset K_{2}\subset K_{3}\subset \cdots \subset \mathrm {relint} (A)} .

Proof

Since we can always go down to the affine span of A {\displaystyle A} , WLOG, the relative interior has dimension n {\displaystyle n} . Now let K j ≡ [ − j , j ] n ∩ { x ∈ int ( K ) : d i s t ( x , ( int ( K ) ) c ) ≥ 1 j } {\displaystyle K_{j}\equiv [-j,j]^{n}\cap \left\{x\in {\text{int}}(K):\mathrm {dist} (x,({\text{int}}(K))^{c})\geq {\frac {1}{j}}\right\}} .

Theorem4—Here "+" denotes Minkowski sum.

  • r e l i n t ( S 1 ) + r e l i n t ( S 2 ) ⊂ r e l i n t ( S 1 + S 2 ) {\displaystyle \mathrm {relint} (S_{1})+\mathrm {relint} (S_{2})\subset \mathrm {relint} (S_{1}+S_{2})} for general sets. They are equal if both S 1 , S 2 {\displaystyle S_{1},S_{2}} are also convex.
  • If S 1 , S 2 {\displaystyle S_{1},S_{2}} are convex and relatively open sets, then S 1 + S 2 {\displaystyle S_{1}+S_{2}} is convex and relatively open.

Theorem5—Here C o n e {\displaystyle \mathrm {Cone} } denotes positive cone. That is, C o n e ( S ) = { r x : x ∈ S , r > 0 } {\displaystyle \mathrm {Cone} (S)=\{rx:x\in S,r>0\}} .

  • C o n e ( r e l i n t ( S ) ) ⊂ r e l i n t ( C o n e ( S ) ) {\displaystyle \mathrm {Cone} (\mathrm {relint} (S))\subset \mathrm {relint} (\mathrm {Cone} (S))} . They are equal if S {\displaystyle S} is convex.
See also

See also

References

References

  1. Zălinescu 2002, pp. 2–3.
  2. Rockafellar, R. Tyrrell (1997) [First published 1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 47. ISBN 978-0-691-01586-6.
  3. Dimitri Bertsekas (1999). Nonlinear Programming (2nd ed.). Belmont, Massachusetts: Athena Scientific. p. 697. ISBN 978-1-886529-14-4.
  4. Rockafellar, R. Tyrrell (1997) [First published 1970]. Convex Analysis. Princeton, NJ: Princeton University Press. Corollary 6.6.2. ISBN 978-0-691-01586-6.
  5. Rockafellar, R. Tyrrell (1997) [First published 1970]. Convex Analysis. Princeton, NJ: Princeton University Press. Theorem 6.9. ISBN 978-0-691-01586-6.
Further reading

Further reading