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 $\text{relint}(S)$ ) is defined as its interior within the affine hull of S.^[1] In other words,

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

where $\text{aff}(S)$ is the affine hull of S, and $N_\epsilon(x)$ 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.

For convex sets C the relative interior can be defined:

$\mathrm{relint}(C)�:= \{x \in C�: \forall_{y \in C} \exists_{z \in C} \exist_{\lambda \in ]0,1[} x=\lambda y %2B (1-\lambda) z\}$

References

^ Zălinescu, C.. Convex analysis in general vector spaces. World Scientific Publishing Co., Inc. pp. 2–3. ISBN 981-238-067-1. MR 1921556.

Boyd, Stephen; Lieven Vandenberghe (2004). Convex Optimization. Cambridge: Cambridge University Press. p. 23. ISBN 0 521 83378 7. http://www.stanford.edu/~boyd/cvxbook/.

Relative interior

See also

References