# Supporting functional

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

## Relation to supporting hyperplane

If ${\displaystyle \phi }$ is a supporting functional of the convex set C at the point ${\displaystyle x_{0}\in C}$ such that

${\displaystyle \phi \left(x_{0}\right)=\sigma =\sup _{x\in C}\phi (x)>\inf _{x\in C}\phi (x)}$

then ${\displaystyle H=\phi ^{-1}(\sigma )}$ defines a supporting hyperplane to C at ${\displaystyle x_{0}}$.[2]

## References

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