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

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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.