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. Intuitively, the relative interior of a set contains all points which are not on the "edge" of the set, relative to the smallest subspace in which this set lies.

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

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

where $\operatorname {aff}(S)$ is the affine hull of S, and $N_{\epsilon }(x)$ 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.

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

\operatorname {relint}(C):=\{x\in C:\forall {y\in C}\;\exists {\lambda >1}:\lambda x+(1-\lambda )y\in C\}.

^[2]^[3]

References

↑ 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.
↑ Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 47. ISBN 978-0-691-01586-6.
↑ Dimitri Bertsekas (1999). Nonlinear Programming (2 ed.). Belmont, Massachusetts: Athena Scientific. p. 697. ISBN 978-1-886529-14-4.

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

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Relative interior

See also

References