| Résume | We prove that for any log-concave random vector X in \mathbb{R}^n with mean zero and identity covariance, \mathbb{E} (|X| - \sqrt{n})^2 \leq C where C > 0 is a universal constant. Thus, most of the mass of the random vector X is concentrated in a thin spherical shell, whose width is only C / \sqrt{n} times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of X related to Eldan's stochastic localization and to the theory of non-linear filtering. A crucial ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture.
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