Optimal Weak Parallelogram Constants for $L^p$ Spaces

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, $1 < p < \infty$.