T-function

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 $\mathrm{relint}(S)$) is defined as its interior within the affine hull of S. In other words,

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

where $\mathrm{aff}(S)$ is the affine hull of S, and $N_{ϵ}(x)$ is a ball of radius $ϵ$ 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 $C\subseteq {ℝ}^n$ the relative interior can be defined as

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

See also

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

References

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