In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.
Let X be a locally convex topological space, and C ⊂ X be a convex set, then the continuous linear functional ϕ : X → R is a supporting functional of C at the point x 0 if ϕ ( x ) ≤ ϕ ( x 0 ) for every x ∈ C .
If h C : X ∗ → R (where X ∗ is the dual space of X ) is a support function of the set C, then if h C ( x ∗ ) = x ∗ ( x 0 ) , it follows that h C defines a supporting functional ϕ : X → R of C at the point x 0 such that ϕ ( x ) = x ∗ ( x ) for any x ∈ X .
If ϕ is a supporting functional of the convex set C at the point x 0 ∈ C such that
ϕ ( x 0 ) = σ = sup x ∈ C ϕ ( x ) > inf x ∈ C ϕ ( x ) then H = ϕ − 1 ( σ ) defines a supporting hyperplane to C at x 0 .
