Optimal Weak Parallelogram Constants for $L^p$ Spaces

Abstract

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

10 pages