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An open convex set can be separated from a convex set with a strict inequality gap.
Let be a real normed space. Let be nonempty convex sets and assume that is open.
Corollary: There exist (see dual space) and such that
This follows from closed hyperplane separation under an interior condition together with the openness of , which allows a strict inequality on the side.
Let be a vector space over (or ). A norm on is a function such that for all and scalars :
A pair is called a normed vector space.
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 metric space and let .
The set is open if for every there exists such that
where is the open ball in .
Open sets are stable under arbitrary unions and finite intersections (see basic properties of open sets). Complements of open sets are closed.
Let be a normed space over .
The (continuous) dual space is the set of all continuous linear functionals (equivalently, all bounded linear functionals on ).
For and , the duality pairing is
This pairing is the standard language used in separation theorems such as separation by a closed hyperplane and strict separation.
Let be a real normed space and let be nonempty convex sets. Assume that int(Ω₁) and
Theorem: The sets and can be separated by a closed hyperplane; i.e., there exists such that
Context: In vector spaces, the analogous separation uses possibly discontinuous linear functionals (see core-based separation). The interior assumption ensures the separating functional is continuous, using closed level-set continuity.