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.

Formally, the relative interior of a set S (denoted ${\displaystyle \operatorname {relint} (S)}$) is defined as its interior within the affine hull of S.[1] In other words,

${\displaystyle \operatorname {relint} (S):=\{x\in S:\exists \epsilon >0,N_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},}$

where ${\displaystyle \operatorname {aff} (S)}$ is the affine hull of S, and ${\displaystyle N_{\epsilon }(x)}$ is a ball of radius ${\displaystyle \epsilon }$ centered on ${\displaystyle x}$. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.

For any nonempty convex set ${\displaystyle C\subseteq \mathbb {R} ^{n}}$ the relative interior can be defined as

${\displaystyle \operatorname {relint} (C):=\{x\in C:\forall {y\in C}\;\exists {\lambda >1}:\lambda x+(1-\lambda )y\in C\}.}$[2][3]

See also

• Interior (topology)
• Algebraic interior
• Quasi-relative interior

References

1. Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing  Co., Inc. pp. 2–3. ISBN 981-238-067-1. MR 1921556.
2. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 47. ISBN 978-0-691-01586-6.
3. Dimitri Bertsekas (1999). Nonlinear Programming (2 ed.). Belmont, Massachusetts: Athena Scientific. p. 697. ISBN 978-1-886529-14-4.

Further reading

• Boyd, Stephen; Lieven Vandenberghe (2004). Convex Optimization. Cambridge: Cambridge University Press. p. 23. ISBN 0-521-83378-7.