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

Damián Pinasco

Mathematische Nachrichten, 2023, vol. 296, issue 8, 3593-3605

Abstract: We prove that given any set of n unit vectors {vi}i=1n⊂Rn$\lbrace v_i\rbrace _{i=1}^{n}\subset \mathbb {R}^n$, the inequality sup∥x∥Rn=1|⟨x,v1⟩⋯⟨x,vn⟩|≥n−n/2$$\begin{equation*} \hspace*{7pc}\sup \limits _{\Vert x \Vert _{\mathbb {R}^n} =1} \vert \langle x, v_1 \rangle \cdots \langle x, v_n\rangle \vert \ge n^{-n/2} \end{equation*}$$holds for n≤14$n \le 14$. Moreover, the equality is attained if and only if {vi}i=1n$\lbrace v_i\rbrace _{i=1}^{n}$ is an orthonormal system.

Date: 2023
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