The projective space has maximal volume among all toric   K\"ahler-Einstein manifolds
  We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained. 
Robert J. Berman, Bo Berndtsson
math.DG