Knowlpedia

Algebraic Interior (Core)

Points from which every vector-space direction remains in the set for a sufficiently short segment.

Let XX be a real and let Ω⊆X\Omega\subseteq X.

The algebraic interior (or core) of Ω\Omega is

core⁡(Ω):={x∈Ω ∣ ∀v∈X, ∃δ>0 s.t. x+tv∈Ω for all ∣t∣<δ}.\operatorname{core}(\Omega):=\Big\{x\in\Omega \ \Big|\ \forall v\in X,\ \exists \delta>0\ \text{s.t.}\ x+tv\in\Omega\ \text{for all }|t|<\delta\Big\}.
Remarks

When XX is a ,

int⁡(Ω)⊆core⁡(Ω)⊆Ω,\operatorname{int}(\Omega)\subseteq \operatorname{core}(\Omega)\subseteq \Omega,

where int⁡(Ω)\operatorname{int}(\Omega) is the usual . Convexity is not needed for these inclusions.

Examples
  • If Ω\Omega is an open ball in a normed space, then core⁡(Ω)=Ω\operatorname{core}(\Omega)=\Omega.
  • If LL is a proper linear subspace of XX, then core⁡(L)=∅\operatorname{core}(L)=\varnothing: a direction outside LL immediately leaves LL.