Extremal Kaehler-Einstein metric for two-dimensional convex bodies

Abstract

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{\"a}hler-Einstein equation $e^{-\Phi} = \det D^2 \Phi$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 \Phi$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 \Phi$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.