Relative interior

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title: "Relative interior" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["topology"] description: "Generalization of topological interior" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Relative_interior" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Generalization of topological 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. 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 \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 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_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).

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_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

| 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

References

References

1. Rockafellar, R. Tyrrell. (1997). "Convex Analysis". [[Princeton University Press]].
2. Dimitri Bertsekas. (1999). "Nonlinear Programming". Athena Scientific.
3. Rockafellar, R. Tyrrell. (1997). "Convex Analysis". [[Princeton University Press]].
4. Rockafellar, R. Tyrrell. (1997). "Convex Analysis". [[Princeton University Press]].

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