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Is there a characterization of those points $z$ in $\mathbb{R}^d$ that satisfy the following:

$\{z \in \mathbb{R}^d : \langle x,z\rangle \leq \|x\|_{p} \text{ for every } x \in \mathbb{R}^d\}$ where $p \geq 1$ with $\langle x,z \rangle$ stands for the inner product.

I am looking for something like "they are all points whose coordinates satisfy... whose $l_q$ norm is..." if there is any characterization for the set above

Thank you...

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You can use \langle and \rangle to produce $\langle$ and $\rangle$ rather than the standard $<, >$. –  JavaMan Feb 12 '13 at 8:30

1 Answer 1

The linear mapping $\mathbb{R}^d \ni x \mapsto \langle x,z \rangle$ satisfies

$$| \langle x,z \rangle| \leq \|x\|_p \cdot \|z\|_q$$

by Hölder's inequality (where $\frac{1}{p}+\frac{1}{q}=1$). Moreover, one can show that this inequality is sharp. Hence $$\{z \in \mathbb{R}^d; \forall x \in \mathbb{R}^d: \langle x,z \rangle \leq \|x\|_p\} = \{z \in \mathbb{R}^d; \|z\|_q \leq 1\} $$

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