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KLS Conjecture for unconditional convex bodies

Prove the Kannan–Lovász–Simonovits conjecture for the class of unconditional log-concave measures or, equivalently, for convex bodies whose densities are invariant under coordinate reflections.

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Background

Unconditional log-concave measures include many classical examples (e.g., product distributions and unconditional convex bodies). Despite additional symmetry, the KLS conjecture has not been resolved even in this restricted setting.

Partial results provide logarithmic bounds on the Poincaré constant for unconditional log-concave measures, but a universal bound (as conjectured by KLS) remains out of reach.

References

We also remark that as of October 2024, the state of affairs is that the KLS conjecture is still open already in the particular case of unconditional convex bodies.

— Isoperimetric inequalities in high-dimensional convex sets (2406.01324 - Klartag et al., 3 Jun 2024) in Section 5 (Bochner identities and curvature), after Corollary on unconditional vectors