Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture

Abstract

Let $R$ be an $m\times m$ correlation matrix satisfying $R-11^{\mathsf{T}}/m⪰0$, let $X\sim \mathcal{N}(0,R)$, and let $Z_1,\dots ,Z_m$ be independent standard Gaussian random variables. We prove ${\max }_i{X}_i{\le }_{\mathrm{st}}{\max }_i{Z}_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in ${\mathbb{R}}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint. The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\le c1\}>\Phi (c{)}^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at https://github.com/abhmul/weak-simplex-conjecture-lean.

42 pages