# Supporting functional

Updated on
Covid-19

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 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 .

## Relation to support function

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 .

## Relation to supporting hyperplane

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 .

## References

Supporting functional Wikipedia

Similar Topics
That Brennan Girl
Steve Book
Jay Mariotti
Topics