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Does the second Clarkson's inequality hold for any two vector in $\mathbb{R}^N$? That is, for any $p\in(1,2)$ and $z,w\in\mathbb{R}^N$, $$ \left|\frac{z+w}{2}\right|^q+\left|\frac{z-w}{2}\right|^q\leq\left[\frac{1}{2}\left(|z|^p+|w|^p\right)\right]^{\frac{1}{p-1}}. $$ Where $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^N$? $\left(\frac{1}{p}+\frac{1}{q}=1\right)$

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    @user54504 I update my answers.2012-12-28
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    Just a remark: any two vectors in $\mathbb R^N$ lie in a two-dimensional subspace (just consider their span). So it suffices to consider $N=2$, where complex arithmetics may help.2012-12-28

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