  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 the
logarithm of the volume is concave along continuous geodesics in the space of
positively curved metrics on $-K_X$ 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 K\"ahler - Einstein metrics and a generalization of this theorem to
'twisted' K\"ahler-Einstein metrics.
