Hyperplane separation theorem

The hyperplane separation theorem states that disjoint convex sets can, under suitable conditions, lie on opposite sides of a hyperplane defined by a linear functional.

Separating hyperplane: A hyperplane whose defining linear functional places two sets in opposite closed half-spaces. It is the geometric object guaranteed by the theorem.
Convex set: A set containing the entire line segment between any two of its points. Convexity is the structural assumption that makes a separating hyperplane possible.
Convex optimization: Optimization of a convex objective over a convex feasible set. Separating hyperplanes underpin optimality and infeasibility arguments in convex programs.
Strict separation theorem: A separation result placing two sets on opposite sides with a strict inequality between their functional values. It strengthens weak separation but requires stronger conditions, such as compactness of one set.
Support function: A function that assigns to each direction the supremum of a set's linear functional in that direction. Support values express the inequalities that certify separation.
Hyperplane: A codimension-one affine subspace, commonly described as the level set of a nonzero linear functional. The theorem's separator is a hyperplane dividing the ambient space.
Farkas' lemma: A theorem giving an alternative between solvability of a linear inequality system and a certificate of infeasibility. It is a finite-dimensional separation result for systems of linear inequalities.
Supporting hyperplane theorem: A theorem guaranteeing a hyperplane through a boundary point that leaves a convex set in one closed half-space. It concerns contact with one convex set's boundary, rather than separation of two disjoint sets.
Dual cone: The cone of linear functionals that are nonnegative on every vector in a given cone. Dual cones encode which functionals preserve the inequalities needed for separation.
Linear functional: A linear map from a vector space to its scalar field. The theorem formulates separation through inequalities for a linear functional.

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Ellipsoid methodRelated: Separation guarantees the kind of geometric certificate used to cut away infeasible points.