Clarkson inequality

The Clarkson inequality says that for all $f,g\in L^p$, for we have:

$${\parallel \frac{f+g}{2}\parallel }_p^p+{\parallel \frac{f-g}{2}\parallel }_p^p\le \frac{1}{2}\left({\parallel f\parallel }_p^p+{\parallel g\parallel }_p^p\right).$$

The inequality can be used to prove that $L^p$ space is uniformly convex for .

Remark. If , then the Clarkson inequality becomes:

$${\parallel \frac{f+g}{2}\parallel }_p^q+{\parallel \frac{f-g}{2}\parallel }_p^q\le {\left(\frac{1}{2}{\parallel f\parallel }_p^p+\frac{1}{2}{\parallel g\parallel }_p^p\right)}^{\frac{1}{p-1}}$$

.

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

Title	Clarkson inequality
Canonical name	ClarksonInequality
Date of creation	2013-03-22 16:04:59
Last modified on	2013-03-22 16:04:59
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	13
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 28A25