  Inspired by Clarkson's inequalities for $L^p$ and continuing work from
\cite{CR}, this paper computes the optimal constant $C$ in the weak
parallelogram laws $$ \|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r +
\|g\|^r \big), $$ $$ \|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r +
\|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.
