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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.

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probability-statisticsJul 15, 2026Significance 20/100Registry: unreviewed

The Weak Simplex Conjecture

Prior state unknown→proved

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.

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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.

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