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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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    What is a multifunction?2012-12-15
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    You can find in the following link: http://en.wikipedia.org/wiki/Multivalued_function2012-12-16
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    OK, but then the next question is what Lipschitz continuity means for a multifunction. Every continuous single-valued branch is Lipschitz or something? Or such that the map $x \mapsto F(x)$ is Lipschitz w.r.t. Hausdorff metric in the image?2012-12-17
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    $F$ is Lipschitz with respect to the Hausdorff distance.2012-12-17

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