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Taking the core twice gives the same set: core(core(Ω))=core(Ω).
Let be a vector space and let be convex.
Corollary:
Here core denotes the algebraic interior.
This follows from segments from core points stay in the core: once a point is in , small segment perturbations remain inside, so taking the core again does not remove any points.
A vector space over a field is a set with operations (functions) and , and an element , satisfying the following for all and :
Let be a vector space (typically over ). A set is convex if for all and all we have
Context. Equivalently, is convex iff it contains every line segment joining any two of its points. Convexity is the core geometric notion underlying convex analysis and optimization.
Let be a vector space and let be convex.
Proposition: If (see core) and , then
where is the half-open line segment from to .