Definition 13.2.6 (Support Function).label Let $C\suf\R^{n}$ be a closed convex set. The support function of $C$ is
\begin{align*}h_{C}:\R^{n}\to\R\cup\curl{\infty}&&h_{C}(y)\define \SUP{x\in C}\inner{x}{y}\end{align*}
If $C$ is compact then $h_{C}(y)=\MAX{x\in C}\inner{x}{y}$. By homogeneity it suffices to consider the restriction $h_{C}|_{\mathbb{S}^{n-1}}:\mathbb{S}^{n-1}\to\R$. Indeed we can extend $\cl{h_C}:\R^{n}\to\R$
\begin{align*}\cl{h_C}(y)=\begin{cases}\cl{h_C}\parens{\frac{y}{\norm{y}{}}\norm{y}{}}=\norm{y}{}h_{C}\parens{\frac{y}{\norm{y}{}}}&y\neq 0\\ 0&y=0\end{cases}\end{align*}
For a convex body $K$ and $y\neq 0$ the hyperplane
\begin{align*}H_{K}(y)\define \curl{x\in\R^n:\inner{x}{y}=h_K(y)}\end{align*}
is a supporting hyperplane of $K$ and
\begin{align*}H_{K}^{-}(y)\define \curl{x\in\R^n:\inner{x}{y}\leq h_K(y)}\end{align*}
is its supporting half-space. If $\norm{y}{}=1$, then $h_{K}(y)$ is the signed distance from the origin to $H_{K}(y)$; when $h_{K}(y)\geq 0$ it is the usual distance. Hence we can write Theorem 13.2.5 as
\begin{align*}K=\caps{}{y\in \mathbb{S}^{n-1}}\curl{x\in\R^n:\inner{x}{y}\leq h_K(y)}\end{align*}
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