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Log-Sobolev KLS Conjecture (Bizeul)

Prove that the largest log-Sobolev constant D_n over all log-concave probability measures μ on ℝ^n satisfying ‖x·θ‖_{ψ2(μ)} ≤ 1 for every direction θ is bounded by a universal constant independent of n.

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Background

This conjecture is a log-Sobolev analogue of KLS: it proposes that, after normalizing all one-dimensional linear marginals to be sub-Gaussian with unit ψ2-norm, the log-Sobolev constant is universally bounded.

Known results yield D_n = O(n) and, via stochastic localization, D_n = O(n{1/2}), but a dimension-free bound remains conjectural.

References

Conjecture [Log-Sobolev KLS conjecture, Bizeul [bizeul_LS]] D_n = O (1) .

— Isoperimetric inequalities in high-dimensional convex sets (2406.01324 - Klartag et al., 3 Jun 2024) in Section 8 (Logarithmic Sobolev inequality and a variant of the KLS conjecture)