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Disjoint convex sets are separable if one has nonempty core and the sets are disjoint.
Let be a real vector space, and let be nonempty convex sets. If
then and can be separated by a hyperplane.
The proof reduces separation of two sets to separation of a point from a convex set by applying point-versus-set separation to the Minkowski difference .
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 real vector space and let be nonempty.
The sets and can be separated by a hyperplane if there is a nonzero linear functional such that
Equivalently,
Any between these two values gives a hyperplane lying between the sets.
Let be a real vector space, let , and let be convex. Assume that core(Ω) is nonempty and .
Theorem. The sets and can be separated by a hyperplane. Equivalently, there exists a nonzero linear functional such that