On the $n-$th linear polarization constant of $\mathbb{R}^n$

Damian Pinasco

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arXiv · 2208.05584

On the $n-$th linear polarization constant of $\mathbb{R}^n$

Damian Pinasco

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Abstract

We prove that given any set of $n$ unit vectors $\{v_i\}_{i=1}^{n}\subset \mathbb R^n,$ the inequality \[ \sup\limits_{\Vert x \Vert_{\mathbb R^n} =1} \vert \langle x, v_1 \rangle \cdots \langle x, v_n\rangle\vert \ge n^{-n/2} \] holds for $n \le 14.$ Moreover, the equality is attained if and only if $\{v_i\}_{i=1}^{n}$ is an orthonormal system.

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BibTeX RIS

Damian Pinasco. 2022-08-10. On the $n-$th linear polarization constant of $\mathbb{R}^n$. https://arxiv.org/abs/2208.05584

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