We show that for any set of $n$ unit vectors $v_1,\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\ldots,p_n$ satisfying $\sum_j p_j = 1$, there exists a unit vector $u$ such that \[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. \] This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequality for products of powers of linear functionals and a strengthening of Bang's classical plank theorem for Hilbert spaces. The proof follows the approach introduced by Martínez and Ortega-Moreno in their recent solution to the strong polarization conjecture posed by Ball and Frenkel. We further note that our weighted inequality admits a Shannon-entropy interpretation: in a random sensing model, the entropy of the weights controls the minimum expected logarithmic loss.

翻译：我们证明：对于实Hilbert空间中的任意一组$n$个单位向量$v_1,\ldots,v_n$及满足$\sum_j p_j = 1$的正数$p_1,\ldots,p_n$，存在一个单位向量$u$使得\[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. \]该不等式是强极化不等式的加权版本。作为直接推论，它导出了线性泛函幂的乘积的极化不等式，以及Hilbert空间中Bang经典板条定理的加强形式。证明方法遵循Martínez与Ortega-Moreno在最近解决Ball和Frenkel提出的强极化猜想时引入的途径。我们进一步指出，该加权不等式具有Shannon熵解释：在随机感知模型中，权重的熵控制了最小期望对数损失。