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If core(Ω1)≠∅ and core(Ω1) is disjoint from Ω2, then Ω1 and Ω2 are separable by a hyperplane.
Let be a real vector space and let be nonempty convex sets. Assume core(Ω₁) and
Theorem: The sets and can be separated by a hyperplane.
Context: This strengthens the disjointness condition by requiring only that avoid the core of . The argument uses idempotence of core and the auxiliary separation lemma.
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 real vector space and let be nonempty.
We say that and can be separated by a hyperplane if there exists a nonzero linear functional such that
Geometrically, picking any with produces the hyperplane lying between the two sets; see the sup/inf characterization.
Let be a real vector space and let be nonempty convex sets. Assume that core(Ω₁) and .
Lemma: The sets and can be separated by a hyperplane.