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

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

where \operatorname{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 any nonempty convex sets C \subseteq \mathbb{R}^n the relative interior can be defined as

\operatorname{relint}(C) := \{x \in C : \forall {y \in C} \; \exists {\lambda > 1}: \lambda x + (1-\lambda)y \in C\}.[2][3]

See also

References

  1. ↑ Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing  Co., Inc. pp. 2–3. ISBN 981-238-067-1. MR 1921556.
  2. ↑ Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 47. ISBN 978-0-691-01586-6.
  3. ↑ Dimitri Bertsekas (1999). Nonlinear Programming (2 ed.). Belmont, Massachusetts: Athena Scientific. p. 697. ISBN 978-1-886529-14-4.