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Thin-Shell implies small-ball deviation via Gaussian tilts
We show that uniform thin-shell estimates for isotropic log-concave measures $μ$ on $\mathbb{R}^n$ 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 t
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