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Litvak's Conjecture on Gaussian Minima

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litvak-gaussian-minima-conjectureProbability & statisticsposed by Alexander Litvak, 2018recorded: disproved

1 attempt · 1 machine check · no person has looked

Statement

Litvak conjectured in 2018 that for every p>0p > 0 the quantity E[min⁡i≤n∣gi∣p]\mathbb{E}[\min_{i \le n} |g_i|^p], for g∼N(0,Σ)g \sim \mathcal{N}(0,\Sigma), is minimized over n×nn \times n correlation matrices by the Gram matrix of the regular simplex in Rn−1\mathbb{R}^{n-1}. False: the matrix Σijcos⁡=cos⁡(π(i−j)/n)\Sigma^{\cos}_{ij} = \cos(\pi(i-j)/n) already gives a strictly smaller value at p=2p = 2, n=4n = 4.

Context

the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture

A named 2018 conjecture on the extremal correlation structure for Gaussian minima, sitting in the same family of extremal questions as the plank and zone problems.

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1 attempt

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  • #1

    Attempt 1

    constructionAlphaEvolve, GPT-5.5 Pro with Dmitriy Kunisky ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    AlphaEvolve, GPT-5.5 Pro
    people
    Dmitriy Kunisky

    Two separate model contributions, and the author keeps them apart. The counterexample itself came out of black-box minimization with AlphaEvolve; what made it usable was the author then recognising a continuous function underneath the numbers and identifying it as the cosine, which turned a numerical optimum into a clean matrix and then into a stronger conjecture. He notes conventional optimizers such as differential evolution and BFGS also produced matrices that would disprove the conjecture, but rarely converged to this one. Separately, GPT-5.5 Pro surfaced the connection to Fejes Toth's zone conjecture, which the author had not known about, while repeatedly producing proofs that leaned on an unsupported step as though it were established. Isolating that step is what produced the volumetric conjecture stated in the paper, so the model's mistake was itself informative.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed site check ·

      scope Reproduction by the VibeMathed site

      Reproduced here. For Σcos⁡\Sigma^{\cos} with n=4n = 4 the matrix has rank two, so gi=Rcos⁡(Θ−πi/4)g_i = R\cos(\Theta - \pi i/4) with R2∼χ22R^2 \sim \chi^2_2, giving the closed form E[min⁡i∣gi∣2]=1−22/π=0.0996836838\mathbb{E}[\min_i |g_i|^2] = 1 - 2\sqrt{2}/\pi = 0.0996836838. The regular-simplex Gram matrix was evaluated by deterministic quadrature over the sphere, stable at 0.14218330.1421833 across grid refinements and corroborated by a 20-million-sample simulation at 0.1422380.142238. The cosine matrix is smaller by about 30 percent, far outside any numerical doubt. arXiv preprint, not peer-reviewed.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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