Optimal Weak Parallelogram Constants for $L^p$ Spaces

TL;DR

This paper determines the exact optimal constants in weak parallelogram laws for $L^p$ spaces, extending Clarkson's inequalities and previous work to provide precise bounds for these fundamental inequalities.

Contribution

It computes the exact optimal constants in weak parallelogram laws for $L^p$ spaces, advancing the understanding of geometric inequalities in functional analysis.

Findings

Calculated the optimal constant C for weak parallelogram laws in $L^p$ spaces.

Extended Clarkson's inequalities to include optimal constants.

Provided precise bounds for weak parallelogram inequalities in $L^p$ 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 $$ \|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big), $$ $$ \|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.