Lemma 13.3.1.label Let $C\suf\R^{n}$ be a nonempty closed convex set and $C\suf\cl{B}(0,R)$ for some $R>0$ then for all $u,v\in\R^{n}$
\begin{align*}\abs{h_C(u)-h_C(v)}\leq R\norm{u-v}{}\end{align*}
In particular, $h_{C}|_{\mathbb{S}^{n-1}}$ is $R$-Lipschitz.
Proof. For every $x\in C$
\begin{align*}\inner{x}{u}\leq\inner{x}{v}+\norm{x}{}\norm{u-v}{}\leq h_{C}(v)+R\norm{u-v}{}\end{align*}
Taking maximum over $x\in C$ gives $h_{C}(u)\leq h_{C}(v)+R\norm{u-v}{}$. Interchanging $u$ and $v$ gives the result.$\square$
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