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
) is defined as its interior within the affine hull of
[1] In other words,
relint (
ϵ > 0
ϵ
{\displaystyle \operatorname {relint} (S):=\{x\in S:{\text{ there exists }}\epsilon >0{\text{ such that }}B_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},}
is the affine hull of
is a ball of radius
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
{\displaystyle \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
the relative interior is equivalently defined as[2][3]
{\displaystyle {\begin{aligned}\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{aligned}}}
means that there exists some
Theorem — If
is nonempty and convex, then its relative interior
is the union of a nested sequence of nonempty compact convex subsets
Since we can always go down to the affine span of
, WLOG, the relative interior has dimension
{\displaystyle K_{j}\equiv [-j,j]^{n}\cap \left\{x\in {\text{int}}(K):\mathrm {dist} (x,({\text{int}}(K))^{c})\geq {\frac {1}{j}}\right\}}
Theorem[4] — Here "+" denotes Minkowski sum.
Theorem[5] — Here
denotes positive cone.
{\displaystyle \mathrm {Cone} (S)=\{rx:x\in S,r>0\}}