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Let $F$ be a Lipschitz continuous multifunction from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the Lipschitz constant $K$ and $H=\sup\{\langle v,p \rangle|\thinspace v \in F(x)\}.$ Prove that $K|p|$ is the Lipschitz constant of $H(.,p)$, i.e,

$|H(x,p)-H(y,p)|\leq K|p||x-y|$ for all $x,y\in\mathbb{R}^n$, for all $p\in\mathbb{R}^n.$

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    $F$ is Lipschitz with respect to the Hausdorff distance.2012-12-17

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