Idempotence of the Core Operator

Taking the core twice gives the same set: core(core(Ω))=core(Ω).

Idempotence of the Core Operator

Let $X$ be a vector space and let $\Omega\subset X$ be convex .

Corollary:

\operatorname{core}(\operatorname{core}(\Omega))=\operatorname{core}(\Omega).

Here core denotes the algebraic interior. This follows from segments from core points stay in the core : once a point is in $\operatorname{core}(\Omega)$ , small segment perturbations remain inside, so taking the core again does not remove any points.