Theorem 13.2.5.label Let $K\suf\R^{n}$ be a convex body then
\begin{align*}K=\caps{}{x_0\in\pd K}H^{-}_{x_0}(a_{x_0},\al_{x_0})=\caps{}{}\curl{H^-:H^-\text{ is a closed supporting half-space of $K$}}\end{align*}
In this sense we say the family of supporting hyperplanes determines $K$ completely.
Proof. The inclusion $\suf$ is immediate. Suppose $y\nin K$ then by Theorem 13.1.25, after choosing orientation of the normal vector, $\exists a\in\R^{n}\del\curl{\vec{0}}$ such that
\begin{align*}\inner{y}{a}>\SUP{x\in K}\inner{x}{a}\end{align*}
By compactness of $K$, the supremum is attained at some $x_{a}\in K$. Hence
\begin{align*}H_{a}\define \curl{x\in\R^n:\inner{x}{a}=\MAX{z\in K}\inner{z}{a}}\end{align*}
is a supporting hyperplane of $K$ and $y\nin H^{-}_{a}=\curl{x\in\R^n:\inner{x}{a}\leq\MAX{z\in K}\inner{z}{a}}$ as desired.$\square$
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