On the n-th linear polarization constant of Rn$mathbb {R}^n$期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Damián Pinasco

Affiliation:	Departamento de Matemáticas y Estadística, Universidad Torcuato Di Tella, Buenos Aires, Argentina

Abstract:	We prove that given any set of n unit vectors ${v_{i}}_{i = 1}^{n} \subset R^{n} $lbrace v_irbrace _{i=1}^{n}subset mathbb {R}^n$$ , the inequality $sup_{{∥ x ∥}_{R^{n}} = 1} \| ⟨ x, v_{1} ⟩ \dots ⟨ x, v_{n} ⟩ \| \geq n^{- n / 2} $$begin{equation} hspace{7pc}sup limits _{Vert x Vert _{mathbb {R}^n} =1} vert langle x, v_1 rangle cdots langle x, v_nrangle vert ge n^{-n/2} end{equation*}$$$ holds for $n \leq 14 $n le 14$$ . Moreover, the equality is attained if and only if ${v_{i}}_{i = 1}^{n} $lbrace v_irbrace _{i=1}^{n}$$ is an orthonormal system.

Keywords:	linear polarization constants product of linear functionals