= Failure of Minkowski inequality below exponent one
{title2=$0<p<1:\quad Q_p((1,1))=2^{1/p}>2$}
For $0<p<1$, write $Q_p(f)=(\int|f|^p\,d\mu)^{1/p}$ to emphasize that this is a <quasi-norm>, as in the <Lq quasi-norm> branch. On two coordinates with <counting measure>, $u=(1,0)$ and $v=(0,1)$ give $Q_p(u)=Q_p(v)=1$, but $Q_p(u+v)=2^{1/p}>2$. At $p=1/2$, this is the explicit failure $4>1+1$ in the right panel of the diagram. The midpoint $(1/2,1/2)$ has $Q_p=2^{1/p-1}>1$, so the corresponding unit sublevel <set> is not a <convex set>.
A useful replacement survives: for nonnegative <real numbers> $a,b$, $(a+b)^p\leq a^p+b^p$. Together with the scalar <triangle inequality>, this yields
$$
Q_p(f+g)^p\leq Q_p(f)^p+Q_p(g)^p.
$$
Consequently $d_p(f,g)=Q_p(f-g)^p$ is a <metric> on the <equivalence classes> of <measurable functions> with finite $Q_p$, even though $Q_p(f-g)$ itself need not be a <metric>. Do not replace the failed <Minkowski inequality> by a universal reversed bound for signed or complex <functions>: $g=-f\ne0$ would give a zero sum but two positive separate sizes.
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