A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in K\"ahler geometry

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