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For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function

$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$

a convex function of $p$ on $(a,b)$ for all $f\in L^1(\mu) \cap L^\infty(\mu)$? When is it concave?

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    Then this is not a norm. More importantly: what did you try?2012-08-15

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