Definitions

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

$text\left\{relint\right\}\left(S\right) = \left\{ x in S : exists epsilon > 0, \left(N_epsilon\left(x\right) cap text\left\{aff\right\}\left(S\right)\right) subset S \right\},$
where $text\left\{aff\right\}\left(S\right)$ is the affine hull of S, and $N_epsilon\left(x\right)$ 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.

## References

*

Search another word or see Relative interioron Dictionary | Thesaurus |Spanish