Litvak’s Gaussian-minima conjecture
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Among centered Gaussian vectors with prescribed unit coordinate variances, which correlation matrix minimizes the expected squared minimum absolute coordinate? The focused Litvak test has n=4 and p=2, comparing the regular simplex with Σᵢⱼ=cos(π(i−j)/n).
Why it matters and what counts as progress
Why it matters: It tests whether the symmetric simplex is a universal minimizer or a different correlation geometry does better.
What counts as progress: A rigorously certified Gaussian correlation matrix beating the simplex value, or a correct minimizer classification for a restricted parameter range.
How the problem developed
- 2018 · Conjecture [1]
Alexander Litvak Conjectured that the regular-simplex correlation matrix minimizes the p-th moment of the smallest absolute coordinate of a centered Gaussian vector, for every p>0.
- 3 May 2026 · Counterexample and revision [2]
Dmitriy Kunisky, with AlphaEvolve-assisted search Disproved the conjecture at p=2 and n=4 using a cosine correlation matrix, then conjectured that this matrix is the minimizer in all dimensions and for all p>0.
AI contribution
A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture
For n=4 and p=2, the AlphaEvolve-found cosine correlation matrix Σᵢⱼ=cos(π(i−j)/n) has a strictly smaller expected squared minimum absolute Gaussian coordinate than the regular-simplex correlation matrix.
Sources
Website timeline
Publication dates and editorial changes.
AlphaEvolve candidate refutes Litvak’s Gaussian-minima conjecture