 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 $1<p<\infty$.

Clarkson's inequalities are a set of mathematical inequalities that relate to norms in functional spaces, particularly in the context of \( L^p \) spaces. They describe how the \( L^p \) norm of sums of functions behaves in relation to the norms of the individual functions.