A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem

For $ϕ$ a metric on the anticanonical bundle, -KX, of a Fano manifold X we consider the volume of X ∫X e-ϕ. We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on -KX and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X. As consequences we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kähler - Einstein metrics and a generalization of this theorem to 'twisted' Kähler-Einstein metrics.