Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2606.02567 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911742463836160 |
|---|---|
| author | Galicer, Daniel Ortega-Moreno, Oscar Pinasco, Damián |
| author_facet | Galicer, Daniel Ortega-Moreno, Oscar Pinasco, Damián |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02567 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong Polarization and Entropy Galicer, Daniel Ortega-Moreno, Oscar Pinasco, Damián Functional Analysis Information Theory 46-xx 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. |
| title | Strong Polarization and Entropy |
| topic | Functional Analysis Information Theory 46-xx |
| url | https://arxiv.org/abs/2606.02567 |