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Let $K=[0,1]\times [0,1]$. Find a continuous mapping $F:K\rightarrow \mathbb{R}^2$ satisfying:

  • $\|F(x)-F(y)\|\leq \|x-y\| \quad\forall x,y\in K,$

  • There exists $\gamma>0$ such that for all $x,y\in K$ $\left\geq 0\Longrightarrow\left\geq \gamma\|x-y\|^2,$

  • There exist $u,v\in K$ such that $\left<0$.

Here, $\|.\|$ is the Euclidean norm and $\left<,.,\right>$ is the scalar product in $\mathbb{R}^2$.

Thank you for all comments and helping.

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    @Robert Israel: Dear Sir. I would like to ask your opinion about this problem.2012-07-24

0 Answers 0