Slicing conjecture (weak form) for convex bodies

Determine whether there exists a universal constant C, independent of n and K, such that the isotropic constant L_K ≤ C for every convex body K ⊂ ℝ^n.

Background

The slicing (or hyperplane) conjecture posits a dimension-free upper bound on the isotropic constant of convex bodies. Its strong form asserts a specific maximizer (the regular simplex), but the classical weak form asks only for the existence of a universal bound across all dimensions and convex bodies.

This conjecture plays a central role in connecting geometric inequalities (like Mahler’s conjecture) via isotropic constants.

References

The usual slicing conjecture postulates that there is a uniform upper bound of the isotropic constant of any convex body in any dimension, while its strong version asserts that, in any fixed dimension, the simplex maximizes the isotropic constant among convex bodies.

— Entropy, slicing problem and functional Mahler's conjecture (2406.07406 - Fradelizi et al., 11 Jun 2024) in Section 1 (Introduction)