  Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex
body whose barycenter is its only interior lattice point. Recently, Berman and
Berndtsson proved this conjecture for a class of rational polytopes including
reflexive polytopes. In particular, they showed that the complex projective
space has the maximal anticanonical degree among all toric Kaehler-Einstein
Fano manifolds. In this note, we prove that projective space is the only such
toric manifold with maximal degree by proving its corresponding
convex-geometric statement. We also discuss a generalized version of Ehrhart's
conjecture involving an invariant corresponding to the so-called greatest lower
bound on the Ricci curvature.

Combinatorics;Differential Geometry;Metric Geometry