= Dual-exponent Clarkson inequality
{title2=$[(\|f+g\|_p^p+\|f-g\|_p^p)/2]^{1/p}\leq(\|f\|_p^{p'}+\|g\|_p^{p'})^{1/p'}$}
For $p\geq2$ and $p'=p/(p-1)$, interpolate the two-coordinate map $(x,y)\mapsto(x+y,x-y)$ between $\ell^1\to\ell^\infty$ <norm> one and $\ell^2\to\ell^2$ <norm> $\sqrt2$. This proves the scalar inequality with factor $2^{1/p}$. Integrating that estimate and applying <Minkowski inequality> to $|f|^{p'}+|g|^{p'}$ in $L^{p/p'}$ gives the displayed function inequality.
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