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If a is in core(Ω) and b in Ω, then points on [a,b) remain in core(Ω).
Let be a vector space and let be convex.
Proposition: If (see core) and , then
where is the half-open line segment from to .
Context: This is the "core" analogue of the interior segment lemma for normed spaces. It is used to prove structural properties of such as idempotence.
A vector space over a field is a set equipped with two operations (functions) and such that for all and :
Vector spaces are the basic objects studied via linear maps and invariants such as the determinant and eigenvalues of operators.
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 .
Lemma. Let be a normed vector space and let be a convex set with nonempty interior. If and , then
where is the half-open segment from to .