Relative interior explained

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 (denoted \operatorname{relint}(S) ) is defined as its interior within the affine hull of S. In other words,$$\backslash operatorname(S)\; :=\; \backslash ,$$where \operatorname{aff}(S) is the affine hull of S, and B _{\epsilon(x)} is a ball of radius \epsilon centered on 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

\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.C \subseteq R ^{n} the relative interior is equivalently defined as^{[1] [2]} where x\in (y,z) means that there exists some 0 < λ < 1 such that x =λ z + ( 1 - λ ) 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.

Further reading

• Book: Convex Optimization. Boyd. Stephen. Lieven Vandenberghe. 2004. Cambridge University Press. Cambridge. 23. 0-521-83378-7.

Notes and References

1. Book: Rockafellar, R. Tyrrell . Convex Analysis . . 1997 . 978-0-691-01586-6 . Princeton, NJ . 47 . Rockafellar, R. Tyrrell . First published 1970.
2. Book: Dimitri Bertsekas . Nonlinear Programming . Athena Scientific . 1999 . 978-1-886529-14-4 . 2nd . Belmont, Massachusetts . 697 . Dimitri Bertsekas.