Strong Polarization and Entropy
2026-06-01 • Information Theory
Information Theory
AI summaryⓘ
The authors prove a new inequality involving a set of unit vectors and positive weights that add up to one, showing there is a specific unit vector meeting a certain sum condition. This result is a weighted version of an existing mathematical inequality related to vector projections. Their finding also improves a classic geometrical result known as Bang's plank theorem in Hilbert spaces. Additionally, they connect their inequality to concepts in information theory, interpreting it in terms of Shannon entropy and expected loss in a random sensing context.
unit vectorsHilbert spacepolarization inequalityBang's plank theoremlinear functionalsweighted inequalitiesShannon entropylogarithmic lossrandom sensingconvex geometry
Authors
Daniel Galicer, Oscar Ortega-Moreno, Damián Pinasco
Abstract
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.