Supporting functional

If ${\displaystyle h_{C}:X^{*}\to \mathbb {R} }$ (where ${\displaystyle X^{*}}$ is the dual space of ${\displaystyle X}$) is a support function of the set C, then if ${\displaystyle h_{C}\left(x^{*}\right)=x^{*}\left(x_{0}\right)}$, it follows that ${\displaystyle h_{C}}$ defines a supporting functional ${\displaystyle \phi :X\to \mathbb {R} }$ of C at the point ${\displaystyle x_{0}}$ such that ${\displaystyle \phi (x)=x^{*}(x)}$ for any ${\displaystyle x\in X}$.

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]