Clarkson's inequalities

For $2\le p<\infty$, the Clarkson inequality isFor $1<p\le 2$ and $q=p/(p-1)$, the other Clarkson inequality isTogether these inequalities prove uniform convexity of $L^p$ for $1<p<\infty$.

For $p\ge 2$ and $p^{\prime }=p/(p-1)$, 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 $\sqrt{2}$. This proves the scalar inequality with factor $2^{1/p}$. Integrating that estimate and applying Minkowski inequality to $|f{|}^{p^{\prime }}+|g{|}^{p^{\prime }}$ in $L^{p/p^{\prime }}$ gives the displayed function inequality.