Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

Given a convex body $$K \subset {\mathbb {R}}^n$$ with the barycenter at the origin, we consider the corresponding Kahler–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 an arbitrary convex body $$K \subseteq {\mathbb {R}}^n$$ is uniformly bounded from above by $$\frac{n-1}{4(n+1)}$$ and we verify this conjecture in the two-dimensional case. The general case remains open.