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The strong isotropic constant conjecture: simplices maximize the isotropic constant

For a convex body K⊂RnK\subset\mathbb R^n with covariance matrix ΣK\Sigma_K of the uniform distribution, the isotropic constant is LK=(det⁡ΣK/∣K∣2)1/(2n)L_K=(\det\Sigma_K/|K|^2)^{1/(2n)}, an affine invariant. Bourgain's slicing problem asked for a dimension-free upper bound, recently obtained by Klartag and Lehec. The sharp (strong) form asks for the extremal body: Campi, Colesanti and Gronchi proved it in the plane, Saroglou proved uniqueness there, and Rademacher showed a simplicial polytope maximizer must be a simplex. Klartag showed the sharp bound implies the non-symmetric Mahler inequality, and Fradelizi and Marin Sola recast it as a sharp entropy bound for log-concave densities. Is it true in every dimension nn that LK≤LΔnL_K\le L_{\Delta_n} for every convex body KK, where Δn\Delta_n is a simplex, with equality only for simplices?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Convex geometry; isotropic constants and the slicing problem
Posed by
Sharp form of Bourgain's slicing problem (1986); no originator named in the manuscript, which cites Campi-Colesanti-Gronchi (1999, planar case) and Fradelizi-Marin Sola (entropy form)
Year posed
—
Years open
—
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1.1: for every n≥1n\ge1 and every convex body K⊂RnK\subset\mathbb R^n, LK≤(n!)1/n/((n+1)(n+1)/(2n)n+2)L_K\le(n!)^{1/n}/((n+1)^{(n+1)/(2n)}\sqrt{n+2}), with equality if and only if KK is a simplex. Also claims the sharp entropy bound h(f)≥m+12log⁡det⁡Cov(f)h(f)\ge m+\tfrac12\log\det\mathrm{Cov}(f) for every log-concave density on Rm\mathbb R^m, with equality exactly for affine images of products of one-sided exponentials, proved via Gaussian-to-target Brenier transport. Via Klartag (2018) it yields the non-symmetric Mahler inequality ∣K∣∣K∘∣≥(n+1)n+1/(n!)2|K||K^\circ|\ge(n+1)^{n+1}/(n!)^2, already listed from a separate manuscript of the same release. It gives no new information on the symmetric slicing constant or on symmetric Mahler.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorems 1.1 and 1.2, read against the conjecture as the manuscript states it. The proof was not refereed. No Lean formalization exists for this family at the pinned commit (no lean/docs/101.md, no comparator challenge). The release ships a scalar certificate checker, verification/verify_scalar.py (exact rational arithmetic, standard library); it was run here on 7 October 2026 and ended with ALL CHECKS PASSED. That checker covers only the rational scalar inequalities used in the estimates, not the transport, chaos-decomposition or equality-case arguments, so it does not raise the tier.

Sources

Changelog1 change

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