Extremal Kahler-Einstein Metric for Two-Dimensional Convex Bodies

Given a convex body K Rn with the barycenter at the origin, we consider the corresponding Kahler-Einstein equation e-phi=detD2 phi. If K is a simplex, then the Ricci tensor of the Hessian metric D2 phi is constant and equals . We conjecture that the Ricci tensor of D2 phi for an arbitrary convex body KRn is uniformly bounded from above by and we verify this conjecture in the two-dimensional case. The general case remains open.