Thin-Shell implies small-ball deviation via Gaussian tilts
Abstract
We show that uniform thin-shell estimates for isotropic log-concave measures μ on ℝn yield precise and explicit deviation estimates for the Euclidean norm |X| below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: μ(|X|≤ √n−s) ≤ exp{ −s2/4 (1+O(s/√n)) } ∀ s∈(0,√n). In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when μ has general covariance structure) of these deviation estimates is also obtained.
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BibTeX
@misc{brazitikos2026thinshell,
title = {{Thin-Shell implies small-ball deviation via Gaussian tilts}},
author = {Silouanos Brazitikos and Emanuel Milman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20816},
eprint = {2608.20816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20816},
}
RIS
TY - UNPB
TI - Thin-Shell implies small-ball deviation via Gaussian tilts
AU - Silouanos Brazitikos
AU - Emanuel Milman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.20816
UR - https://arxiv.org/abs/2608.20816
AB - We show that uniform thin-shell estimates for isotropic log-concave measures μ on ℝ^n yield precise and explicit deviation estimates for the Euclidean norm |X| below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: μ(|X|≤ √n−s) ≤ exp{ −s^2/4 (1+O(s/√n)) } ∀ s∈(0,√n). In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when μ has general covariance structure) of these deviation estimates is also obtained.
ER -