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 $$\mathrm{\parallel }f+g{\mathrm{\parallel }}^r+C\mathrm{\parallel }f-g{\mathrm{\parallel }}^r\le 2^{r-1}(\mathrm{\parallel }f{\mathrm{\parallel }}^r+\mathrm{\parallel }g{\mathrm{\parallel }}^r),$$ $$\mathrm{\parallel }f+g{\mathrm{\parallel }}^r+C\mathrm{\parallel }f-g{\mathrm{\parallel }}^r\ge 2^{r-1}(\mathrm{\parallel }f{\mathrm{\parallel }}^r+\mathrm{\parallel }g{\mathrm{\parallel }}^r)$$ for the $L^p$ spaces, $1<p<\mathrm{\infty }$.