Definition 13.2.1 (Supporting Hyperplanes).label Let $C\suf\R^{n}$ be a convex set, $C\neq\emp,\R^{n}$. We call a hyperplane $H=\curl{x\in\R^n:\inner{x}{a}=\al}$ a supporting hyperplane if
\begin{align*}C\suf H^{-}\define \curl{x\in R^n:\inner{x}{a}\leq \al}&&\pd C\cap H\neq\emp\end{align*}
Equivalently we can require $\cl{C}\cap H\neq\emp$. The closed half-space $H^{-}$ is called the supporting half-space.
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