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:

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

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