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

We prove a stochastic comparison for Gaussian maxima. Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1} \mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. Then $\max_{1\leq i\leq m}X_i \leq_{\mathrm{st}} \max_{1\leq i\leq m}Z_i$, or equivalently, $\mathbb{P}\{X_i\leq c\text{ for every }i\}\geq\Phi(c)^m$ for every $c\in\mathbb{R}$. This comparison resolves 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 maximizes the probability of correct maximum-likelihood decoding at every signal-to-noise ratio. It also proves the inequality asserted by the Simplex Mean Width Conjecture and gives an exact formula for the largest number of equiprobable messages that can be sent at prescribed energy and error probability by a deterministic no-feedback AWGN code under a per-codeword energy constraint. The proof combines a Gaussian product inequality for log-concave functions with an adaptive tilting argument that makes the inequality applicable to the one-sided threshold events defining the maximum.