Vector balancing in convex order
Abstract
We construct Komlós signing laws with discrepancy below $6.84$, independent Gaussian reference blocks and exponentially many balanced signings. One law preserves hard constraints and exact conditional means while its reference controls every joint convex cost. We resolve both Reis--Rothvoss Schatten conjectures. For n symmetric n× n matrices, one prescribed-mean signing law bounds all centered Schatten discrepancies with sharp powers of n and a universal constant. One Gaussian event of measure e−O(n) controls every unitarily invariant norm at the square-function radius. We prove Reis's volume-ratio conjecture: every r-dimensional quotient of the a× b trace-class unit ball has volume ratio O(√(1+min(a,b)/r)). We resolve Nutz--Wang--Zhang's directional and supermartingale conjectures and the higher-dimensional Bass-flow conjecture. For square-integrable laws, full convex reference support is the universal finite-potential criterion in Tschiderer's q-Bass problem. Every driver, even atomic, admits one extended potential and shift for all maximal-covariance martingale optimizers of each irreducible pair with full terminal affine span. For Talagrand's problem, five copies of a compact balanced Gaussian half-mass set contain compact convex output of mass at least 3/4, also in Wiener space. Cheeger's bound is asymptotically exact throughout subexponential Dirichlet spectral ranges of mixed weighted products. For standard Borel sources, a Blackwell-greatest perfectly private full-data release exists exactly when, after discarding conditional point masses, the rest coincide or share a two-point support. Under conditional atomlessness, existence is equivalent to independence. Covariance-matched analytic uniformly log-concave sources can have a greatest useful-data-only private release with infinite information, while no greatest full-data release exists.
Related
BibTeX
@misc{ercan2026vector,
title = {{Vector balancing in convex order}},
author = {Eren Ercan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38228},
eprint = {2609.38228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38228},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vector balancing in convex order
AU - Eren Ercan
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.38228
UR - https://arxiv.org/abs/2609.38228
AB - We construct Komlós signing laws with discrepancy below $6.84$, independent Gaussian reference blocks and exponentially many balanced signings. One law preserves hard constraints and exact conditional means while its reference controls every joint convex cost. We resolve both Reis--Rothvoss Schatten conjectures. For n symmetric n× n matrices, one prescribed-mean signing law bounds all centered Schatten discrepancies with sharp powers of n and a universal constant. One Gaussian event of measure e^−O(n) controls every unitarily invariant norm at the square-function radius. We prove Reis's volume-ratio conjecture: every r-dimensional quotient of the a× b trace-class unit ball has volume ratio O(√(1+min(a,b)/r)). We resolve Nutz--Wang--Zhang's directional and supermartingale conjectures and the higher-dimensional Bass-flow conjecture. For square-integrable laws, full convex reference support is the universal finite-potential criterion in Tschiderer's q-Bass problem. Every driver, even atomic, admits one extended potential and shift for all maximal-covariance martingale optimizers of each irreducible pair with full terminal affine span. For Talagrand's problem, five copies of a compact balanced Gaussian half-mass set contain compact convex output of mass at least 3/4, also in Wiener space. Cheeger's bound is asymptotically exact throughout subexponential Dirichlet spectral ranges of mixed weighted products. For standard Borel sources, a Blackwell-greatest perfectly private full-data release exists exactly when, after discarding conditional point masses, the rest coincide or share a two-point support. Under conditional atomlessness, existence is equivalent to independence. Covariance-matched analytic uniformly log-concave sources can have a greatest useful-data-only private release with infinite information, while no greatest full-data release exists.
ER -