Strong isotropic constant conjecture (simplex maximizes)

Prove that for every convex body K ⊂ R^n, the isotropic constant satisfies L_K ≤ L_{Δ_n}, where Δ_n denotes an n-dimensional simplex. This would identify simplices as global maximizers of the isotropic constant among all convex bodies in R^n.

Background

Beyond the existence of a universal bound, a stronger assertion posits that the extremal bodies are simplices in every dimension. The paper records this stronger conjecture and studies structural constraints on putative polytopal maximizers that provide supporting evidence.

Rademacher’s theorem and the new results in the paper show that under certain local maximizing assumptions and boundary structures (e.g., simplicial vertices or affine reflectors), a polytopal local maximizer must be a simplex, aligning with the strong conjecture.