Extremal Kaehler-Einstein metric for two-dimensional convex bodies

TL;DR

This paper investigates the properties of Kähler-Einstein metrics associated with convex bodies, proving a conjecture about Ricci tensor bounds in two dimensions and highlighting the open problem in higher dimensions.

Contribution

It verifies a conjecture that the Ricci tensor of the Hessian metric is bounded for two-dimensional convex bodies, extending understanding of Kähler-Einstein metrics in convex geometry.

Findings

Ricci tensor is constant for simplices

Bound on Ricci tensor verified in 2D case

Open problem for higher dimensions

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.