TITLE: Example when triangle inequality fails in weak $L^p$ spaces
QUESTION [6 upvotes]: Let's consider the quasi-norm on the weak $L^p(X,m)$ spaces: $$[f]_p=\sup_{t>0}\left\{t\, m\big(\{x:|f(x)|>t\}\big)^{1/p}\right\}.$$ We know that it is not a norm since the triangle inequality fails. Can you give me an example? For simplicity we can just consider the case $p=2$.

REPLY [5 votes]: Let $(X,m)$ be the interval $[0,1]$ with the Lebesgue measure. Assume $1<p<\infty$. 
Define $f(x)=x^{-1/p}$ and $g(x)=f(1-x)$. Clearly, $[f]_p=[g]_p=1$. By convexity of $f$, the sum $f+g$ attains its minimum at $x=1/2$. Since this minimum us $2^{1+1/p}$, it follows that $$[f+g]_p\ge 2^{1+1/p}>2=[f]_p+[g]_p$$