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{ä}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.

References in corpus (1)

• Eigenvalue distribution of optimal transportation