Digesting the proof of the sharp thin-shell inequality
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 $X = (X_1,\ldots,X_n)$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ {\rm Var}( |X|^2 ) \leq 8 n. $$ The constant $8$ is optimal: equality is attained when $X_1,\ldots,X_n$ are independent, identically distributed, standard, centered exponential random variables. Moreover, among isotropic random vectors distributed uniformly on convex bodies in $\mathbb{R}^n$, 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 $3^{rd}$-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-Amp\`ere equation. This manifold was studied in this context in \cite{lc_moment}. The main improvement over \cite{lc_moment} comes from a concise yet effective analysis of the $3^{rd}$-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.