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let $F$ a closed convex subset of $\mathbb{R}^n$, let $x,y\in F$ and assume that for any $s\in[0,1]$ we have $f(s):=\mid sx+(1-s)y-z\mid\geq \mid y-z\mid$ why is it true that $\frac{\partial}{\partial s} f(s) |_{s=0}\geq 0$?

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    Maybe because $f(s)\ge f(0)$ for every $s\ge0$?2011-04-18
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    What is $z$ in your expression?2011-04-18

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