Relative interior

From Wikipedia, the free encyclopedia

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

$\text{relint}(S) = \{ x \in S : \exists \epsilon > 0, (N_\epsilon(x) \cap \text{aff}(S)) \subset S \},$

where $aff(S)$ 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.

[edit] References

Boyd, Stephen; Lieven Vandenberghe (2004). Convex Optimization. Cambridge: Cambridge University Press, p. 23. ISBN 0 521 83378 7.

Relative interior

From Wikipedia, the free encyclopedia

[edit] References

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