2026/07/25 by Yuansi Chen, Boaz Klartag
#math.MG #math.FA #math.PR
We present a proof that determines the optimal value of the universal constant in the thin-shell theorem for log-concave distributions in high dimensions. We prove that for any log-concave random vector X = (X1,…,Xn) in ℝn with mean zero and identity covariance, \rm Var( |X|2 ) ≤ 8 n. The constant 8 is optimal: equality is attained when X1,…,Xn are independent, identically distributed, standard, centered exponential random variables. Moreover, among isotropic random vectors distributed uniformly on convex bodies in ℝn, the quantity \rm Var(|X|2) is maximized by the uniform distribution on a regular simplex. We also provide a corresponding sharp bound on the Hilbert-Schmidt norm of the tensor of 3rd-moments of isotropic, log-concave distributions. The argument relies on the analysis of a weighted Riemannian manifold associated with log-concave moment measures and the Monge-Ampère equation. This manifold was studied in this context in \citelcmoment. The main improvement over \citelcmoment comes from a concise yet effective analysis of the 3rd-derivatives tensor of the potential. The proof was found by GPT-5.6 Pro in response to prompts supplied by the first-named author, following general discussions between the two authors concerning log-concave moment measures. The prompts referred to the paper ``Logarithmically-concave moment measures I'' and suggested bootstrapping a bound on the second trace moment.