Sudakov Minoration for Unconditional Log-Concave Vectors
Abstract
Sudakov minoration asks whether a large family of separated random linear forms must have a large expected maximum. We present a proof candidate for this principle for all unconditional log-concave vectors. The argument starts with a classical reduction to sparse coordinate supports. It then selects one feasible threshold witness for each label, removes a common coordinate core, and controls the remaining overlaps. The main analytic step is an exponential-moment estimate for a softened witness payoff. We obtain it by combining a one-dimensional clipping inequality with triangular transport, using log-concavity of the source and sign symmetry of the target. A Bernoulli comparison restores the signs at the end. The exposition includes historical context, a guide to the proof, all intermediate arguments, and explicit choices of constants. Two consequences concern covering numbers in the moment metric and concentration of bounded functions of the magnitudes.
Related
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- On the Monotonicity of Discrete Entropy for Log-Concave Random Vectors on ℤd
BibTeX
@misc{bednorz2026sudakov,
title = {{Sudakov Minoration for Unconditional Log-Concave Vectors}},
author = {Witold Bednorz and Rafal Martynek and Rafal Meller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31782},
eprint = {2609.31782},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31782},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sudakov Minoration for Unconditional Log-Concave Vectors
AU - Witold Bednorz
AU - Rafal Martynek
AU - Rafal Meller
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.31782
UR - https://arxiv.org/abs/2609.31782
AB - Sudakov minoration asks whether a large family of separated random linear forms must have a large expected maximum. We present a proof candidate for this principle for all unconditional log-concave vectors. The argument starts with a classical reduction to sparse coordinate supports. It then selects one feasible threshold witness for each label, removes a common coordinate core, and controls the remaining overlaps. The main analytic step is an exponential-moment estimate for a softened witness payoff. We obtain it by combining a one-dimensional clipping inequality with triangular transport, using log-concavity of the source and sign symmetry of the target. A Bernoulli comparison restores the signs at the end. The exposition includes historical context, a guide to the proof, all intermediate arguments, and explicit choices of constants. Two consequences concern covering numbers in the moment metric and concentration of bounded functions of the magnitudes.
ER -