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 $\operatorname{relint}(S)$ ) is defined as its interior within the affine hull of $S.$ {{sfn|Zălinescu|2002|pp=2-3}} In other words,

$\operatorname{relint}(S) := \{ x \in S : \text{ there exists } \epsilon > 0 \text{ such that } B_\epsilon(x) \cap \operatorname{aff}(S) \subseteq S \},$

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.

For any convex set $C \subseteq \mathbb{R}^n$ the relative interior is equivalently defined as{{cite book |author=Rockafellar, R. Tyrrell |title=Convex Analysis |publisher=Princeton University Press |year=1997 |isbn=978-0-691-01586-6 |location=Princeton, NJ |page=47 |author-link=Rockafellar, R. Tyrrell |orig-year=First published 1970}}{{cite book |author=Dimitri Bertsekas |title=Nonlinear Programming |publisher=Athena Scientific |year=1999 |isbn=978-1-886529-14-4 |edition=2nd |location=Belmont, Massachusetts |page=697 |author-link=Dimitri Bertsekas}}

$\begin{align}\operatorname{relint}(C) &:= \{x \in C : \text{ for all } y \in C, \text{ there exists some } \lambda > 1 \text{ such that } \lambda x + (1 - \lambda) y \in C\}\\
&= \{x \in C : \text{ for all } y\neq x \in C, \text{ there exists some } z \in C \text{ such that } x\in (y,z)\}.
\end{align}$

where $x\in (y,z)$ means that there exists some $0< \lambda < 1$ such that $x=\lambda z + (1 - \lambda) y$ .

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

{{Math theorem

| math_statement = If $A\subset \R^n$ is nonempty and convex, then its relative interior $\mathrm{relint}(A)$ is the union of a nested sequence of nonempty compact convex subsets $K_1\subset K_2\subset K_3\subset\cdots \subset \mathrm{relint}(A)$ .

}}

{{Math proof|title=Proof|proof=

Since we can always go down to the affine span of $A$ , WLOG, the relative interior has dimension $n$ . Now let $K_j \equiv [-j,j]^n \cap \left\{ x \in \text{int}(K) : \mathrm{dist}(x, (\text{int}(K))^c) \ge \frac{1}{j} \right\}$ .

}}

{{Math theorem

| math_statement = Here "+" denotes Minkowski sum.

$\mathrm{relint}(S_1) + \mathrm{relint}(S_2) \subset \mathrm{relint}(S_1 + S_2)$ for general sets. They are equal if both $S_1, S_2$ are also convex.

If $S_1, S_2$ are convex and relatively open sets, then $S_1 + S_2$ is convex and relatively open.

| name = Theorem{{cite book |author=Rockafellar, R. Tyrrell |title=Convex Analysis |publisher=Princeton University Press |year=1997 |isbn=978-0-691-01586-6 |location=Princeton, NJ |at=Corollary 6.6.2 |author-link=Rockafellar, R. Tyrrell |orig-year=First published 1970}}

}}

{{Math theorem

| math_statement = Here $\mathrm{Cone}$ denotes positive cone. That is, $\mathrm{Cone}(S) = \{rx: x\in S, r > 0\}$ .

$\mathrm{Cone}(\mathrm{relint}(S)) \subset \mathrm{relint}(\mathrm{Cone}(S))$ . They are equal if $S$ is convex.

| name = Theorem{{cite book |author=Rockafellar, R. Tyrrell |title=Convex Analysis |publisher=Princeton University Press |year=1997 |isbn=978-0-691-01586-6 |location=Princeton, NJ |at=Theorem 6.9 |author-link=Rockafellar, R. Tyrrell |orig-year=First published 1970}}

}}

References

{{Zălinescu Convex Analysis in General Vector Spaces 2002}}

Relative interior

Comparison to interior

Properties

See also

References

Further reading