From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality
Abstract
We establish dimension-free, pointwise optimal lower-deviation bounds for convex functions under spherically symmetric Cauchy-type measures. These bounds use either mean-variance normalization or moments about the median, the latter covering the Cauchy case where the mean and variance are unavailable. By approximation with a parametrized family the Cauchy-type measures, our results recover the Gaussian setting of Paouris and Valettas \cite{PaourisValettas18} and strengthen their variance-sensitive estimate to a sharp bound, with a complete equality characterization. This goes beyond the partial improvements obtained in subsequent work. We also prove an Ehrhard-type inequality for arbitrary convex sets under these Cauchy-type measures in every dimension and characterize all equality cases. This resolves a conjecture posed by Byczkowski, Małecki and Żak \cite{ByczkowskiMaleckiZak22}. The proof follows from a variational argument combined with the joint concavity of fiber rearrangement. The geometric inequality reduces distributions of convex observables to one-dimensional convex transforms, yielding the sharp lower-deviation bounds.
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BibTeX
@misc{huang2026from,
title = {{From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality}},
author = {Yong Huang and Shengyu Tang and Sinan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14880},
eprint = {2610.14880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14880},
}
RIS
TY - UNPB
TI - From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality
AU - Yong Huang
AU - Shengyu Tang
AU - Sinan Wang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14880
UR - https://arxiv.org/abs/2610.14880
AB - We establish dimension-free, pointwise optimal lower-deviation bounds for convex functions under spherically symmetric Cauchy-type measures. These bounds use either mean-variance normalization or moments about the median, the latter covering the Cauchy case where the mean and variance are unavailable. By approximation with a parametrized family the Cauchy-type measures, our results recover the Gaussian setting of Paouris and Valettas \cite{PaourisValettas18} and strengthen their variance-sensitive estimate to a sharp bound, with a complete equality characterization. This goes beyond the partial improvements obtained in subsequent work. We also prove an Ehrhard-type inequality for arbitrary convex sets under these Cauchy-type measures in every dimension and characterize all equality cases. This resolves a conjecture posed by Byczkowski, Małecki and Żak \cite{ByczkowskiMaleckiZak22}. The proof follows from a variational argument combined with the joint concavity of fiber rearrangement. The geometric inequality reduces distributions of convex observables to one-dimensional convex transforms, yielding the sharp lower-deviation bounds.
ER -