Relative interior

Generalization of topological 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.

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:{\text{ there exists }}\epsilon >0{\text{ such that }}B_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},

where

\operatorname {aff} (S)

is the affine hull of

S,

and

B_{\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.

A set is relatively open iff it is equal to its relative interior. Note that when $\operatorname {aff} (S)$ is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed.

For any convex set $C\subseteq \mathbb {R} ^{n}$ the relative interior is equivalently defined as^[2]^[3]

{\begin{aligned}\operatorname {relint} (C)&:=\{x\in C:{\text{ for all }}y\in C,{\text{ there exists some }}\lambda >1{\text{ such that }}\lambda x+(1-\lambda )y\in C\}\\&=\{x\in C:{\text{ for all }}y\neq x\in C,{\text{ there exists some }}z\in C{\text{ such that }}x\in (y,z)\}.\end{aligned}}

where

x\in (y,z)

means that there exists some

0<\lambda <1

such that

x=\lambda z+(1-\lambda )y

Comparison to interior

The interior of a point in an at least one-dimensional ambient space is empty, but its relative interior is the point itself.

The interior of a line segment in an at least two-dimensional ambient space is empty, but its relative interior is the line segment without its endpoints.

The interior of a disc in an at least three-dimensional ambient space is empty, but its relative interior is the same disc without its circular edge.

Properties

Theorem — If $A\subset \mathbb {R} ^{n}$ is nonempty and convex, then its relative interior $\mathrm {relint} (A)$ is the union of a nested sequence of nonempty compact convex subsets $K_{1}\subset K_{2}\subset K_{3}\subset \cdots \subset \mathrm {relint} (A)$ .

Proof

Since we can always go down to the affine span of $A$ , WLOG, the relative interior has dimension $n$ . Now let $K_{j}\equiv [-j,j]^{n}\cap \left\{x\in {\text{int}}(K):\mathrm {dist} (x,({\text{int}}(K))^{c})\geq {\frac {1}{j}}\right\}$ .

Theorem^[4] — Here "+" denotes Minkowski sum.

$\mathrm {relint} (S_{1})+\mathrm {relint} (S_{2})\subset \mathrm {relint} (S_{1}+S_{2})$ for general sets. They are equal if both $S_{1},S_{2}$ are also convex.

If $S_{1},S_{2}$ are convex and relatively open sets, then $S_{1}+S_{2}$ is convex and relatively open.

Theorem^[5] — Here $\mathrm {Cone}$ denotes positive cone. That is, $\mathrm {Cone} (S)=\{rx:x\in S,r>0\}$ .

$\mathrm {Cone} (\mathrm {relint} (S))\subset \mathrm {relint} (\mathrm {Cone} (S))$ . They are equal if $S$ is convex.

References

^ Zălinescu 2002, pp. 2–3.
^ Rockafellar, R. Tyrrell (1997) [First published 1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 47. ISBN 978-0-691-01586-6.
^ Dimitri Bertsekas (1999). Nonlinear Programming (2nd ed.). Belmont, Massachusetts: Athena Scientific. p. 697. ISBN 978-1-886529-14-4.
^ Rockafellar, R. Tyrrell (1997) [First published 1970]. Convex Analysis. Princeton, NJ: Princeton University Press. Corollary 6.6.2. ISBN 978-0-691-01586-6.
^ Rockafellar, R. Tyrrell (1997) [First published 1970]. Convex Analysis. Princeton, NJ: Princeton University Press. Theorem 6.9. ISBN 978-0-691-01586-6.

Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category