Universal upper bound for isotropic constants

Determine whether there exists a universal constant C > 0 such that for every dimension n ≥ 1 and every convex body K ⊂ R^n, the isotropic constant satisfies L_K ≤ C. Establishing such a dimension-independent bound would resolve the isotropic constant conjecture.

Background

The isotropic constant L_K is an affine-invariant parameter of a convex body K and plays a central role in high-dimensional convex geometry. Significant partial progress is known: Bourgain, Klartag, Chen, Klartag–Lehec, and Klartag have provided successive upper bounds culminating in L_K ∈ O(√log n).

Despite these advances, no dimension-free upper bound is known, and the existence of such a constant would resolve a longstanding problem. The paper references this as the isotropic constant conjecture and situates its results within this broader context.