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For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold$X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that thelogarithm of the volume is concave along continuous geodesics in the space ofpositively curved metrics on $-K_X$ and that the concavity is strict unless thegeodesic comes from the flow of a holomorphic vector field on $X$. Asconsequences we get a simplified proof of the Bando-Mabuchi uniqueness theoremfor K\"ahler - Einstein metrics and a generalization of this theorem to'twisted' K\"ahler-Einstein metrics.

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