Mahler’s second conjecture (general case)

Prove that for every convex body K ⊂ R^n (without symmetry), the Mahler volume M(K) = n! |K| |K°| satisfies M(K) ≥ (n+1)^{n+1} / n!, with equality attained by simplices whose barycenter is at the origin.

Background

Mahler’s second conjecture concerns the optimal lower bound of the Mahler volume among all convex bodies, not necessarily symmetric. It predicts that simplices (centered at the origin) minimize the volume product and gives the explicit bound (n+1)^{n+1}/n! (asymptotic to e^n by Stirling’s formula).