2025/07/21 by Klartag, Boaz, Lehec, Joseph · 6 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.2507.15495
We prove that for any log-concave random vector X in ℝn with mean zero and identity covariance, 𝔼 (|X| - √(n))2 ≤ 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 / √(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.