Discoveries / probability-statistics

The Weak Simplex Conjecture

Among $d+1$ equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any $m \times m$ correlation matrix $R$ with $R - \mathbf{1}\mathbf{1}^{\mathsf T}/m \succeq 0$ and $X \sim \mathcal{N}(0,R)$, the maximum of the $X_i$ is stochastically dominated by the maximum of $m$ independent standard Gaussians.