Why is the norm a convex function?                          

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    $\begingroup$ The norm of what???? $\endgroup$ – coffeemath Nov 1 '15 at 16:40
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    $\begingroup$ @coffeemath: What do you mean? The norm is a function. Functions can be convex. $\endgroup$ – user541686 Nov 1 '15 at 20:50

By triangle inequality \begin{align} \left\|ax + \left(1-a\right)y \right\|\le \left\|ax \right\| + \left\|\left(1-a\right)y \right\|= a\left\|x \right\| + \left(1-a\right)\left\|y\right\| \end{align}

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