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The Weak Simplex Conjecture

Probability & statistics · posed by Claude Shannon; Balakrishnan; Weber, 1965 · solved

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Statement

Among d+1d+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×mm \times m correlation matrix RR with R−11T/m⪰0R - \mathbf{1}\mathbf{1}^{\mathsf T}/m \succeq 0 and X∼N(0,R)X \sim \mathcal{N}(0,R), the maximum of the XiX_i is stochastically dominated by the maximum of mm independent standard Gaussians.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Abhijeet Mulgund, using GPT-5.6 Pro, GPT-5.6 Sol Max

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    The disclosure says the models were used during the development of the work to assist with mathematical exploration, proof development and checking, literature discovery and organization, and drafting. It attributes no specific step, so the lowest tier applies.

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