# Clarkson inequality

The Clarkson inequality says that for all $f,g\in L^{p}$, for $2\leq p<\infty$ we have:

 $\left\|\frac{f+g}{2}\right\|^{p}_{p}+\left\|\frac{f-g}{2}\right\|^{p}_{p}\leq% \frac{1}{2}\left(\|f\|^{p}_{p}+\|g\|^{p}_{p}\right).$

The inequality can be used to prove that $L^{p}$ space is uniformly convex for $2\leq p<\infty$.

Remark. If $1, then the Clarkson inequality becomes:

 $\left\|\frac{f+g}{2}\right\|^{q}_{p}+\left\|\frac{f-g}{2}\right\|^{q}_{p}\leq% \left(\frac{1}{2}\|f\|^{p}_{p}+\frac{1}{2}\|g\|^{p}_{p}\right)^{\frac{1}{p-1}}$

.

for functions $f,\,g\in L^{p}$, where $q=\frac{p}{p-1}$.

Title Clarkson inequality ClarksonInequality 2013-03-22 16:04:59 2013-03-22 16:04:59 georgiosl (7242) georgiosl (7242) 13 georgiosl (7242) Theorem msc 28A25