Strong slicing problem: simplex attainment and the constant 1/e

Determine whether, for each dimension n, the supremum Ln := sup_{K ⊂ R^n} L_K over isotropic constants of convex bodies is attained by a simplex, and ascertain whether sup_n Ln = 1/e. Here L_K denotes the affine-invariant isotropic constant of the convex body K defined via the determinant of the covariance of the uniform measure on K and its volume.

Background

The paper proves Theorem 1.2, establishing a uniform bound on Ln across all dimensions, thereby resolving Bourgain’s slicing problem in the affirmative. Beyond boundedness, a stronger form asks for an exact extremizer and value: whether the bodies maximizing L_K are simplices and whether the global supremum sup_n Ln equals 1/e.

The authors note that an affirmative resolution of this strong form would imply Mahler’s conjecture concerning the product of the volume of a convex body and the volume of its polar body, highlighting the broader significance of the precise extremal structure.

References

A strong version of the slicing problem asks whether the supremum in (2) is attained when K C R™ is a simplex. If the answer is affirmative and indeed supn Ln = 1/e, this would imply Mahler's conjecture on the product of the volume of a convex body and the volume of its polar body, see [19].

— Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound (2412.15044 - Klartag et al., 19 Dec 2024) in Section 1 (Introduction), paragraph after Theorem 1.2