is product of norms convex? Is a function of the form 
$f(x) = \|x\|_1\|x\|_2$
convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet

Hint: we have $(\lVert x\rVert_1+\lVert y\rVert_1)((\lVert x\rVert_2+\lVert y\rVert_2)\geqslant 0$. Multiply this by $\alpha(1-\alpha)\geqslant 0$. It's the difference between $$\alpha f(x)+(1-\alpha)f(y)\mbox{ and  }\alpha \lVert x\rVert_1+(1-\alpha)\lVert y\rVert_1))(\alpha \lVert x\rVert_2+(1-\alpha)\lVert y\rVert_2).$$
