Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies
Given a convex body K⊂RnK⊂Rn with the barycenter at the origin, we consider the corresponding Kähler–Einstein equation e−Φ=detD2Φe−Φ=detD2Φ. If K is a simplex, then the Ricci tensor of the Hessian metric D2ΦD2Φ is constant and equals n−14(n+1)n−14(n+1). We conjecture that the Ricci tensor of D2ΦD2Φfor an arbitrary convex body K⊆RnK⊆Rn is uniformly bounded from above by n−14(n+1)n−14(n+1) and we verify this conjecture in the two-dimensional case. The general case remains open.