  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 bounded 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 a
consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem
for K\"ahler - Einstein metrics. A generalization of this theorem to 'twisted'
K\"ahler-Einstein metrics and some classes of manifolds that satisfy weaker
hypotheses than being Fano is also given. We moreover discuss a generalization
of the main result to other bundles than $-K_X$, and finally use the same
method to give a new proof of the theorem of Tian and Zhu of uniqueness of
K\"ahler-Ricci solitons. This is an expanded version of an earlier preprint, "A
Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi
uniqueness theorem", arXiv:1103.0923
