Supporting functional
In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set. == Mathematical definition == Let X be a locally convex topological space, and C ⊂ X {\displaystyle C\subset X} be a convex set, then the continuous linear functional ϕ : X → R {\displaystyle \phi :X\to \mathbb {R} } is a supporting functional of C at the point x 0 {\displaystyle x_{0}} if ϕ ≠ 0 {\displaystyle \phi \not =0} and ϕ ( x ) ≤ ϕ ( x 0 ) {\displaystyle \phi (x)\leq \phi (x_{0})} for every x ∈ C {\displaystyle x\in C} .