  We establish the convexity of Mabuchi's K-energy functional along weak
geodesics in the space of Kahler potentials on a compact Kahler manifold thus
confirming a conjecture of Chen and give some applications in Kahler geometry,
including a proof of the uniqueness of constant scalar curvature metrics (or
more generally extremal metrics) modulo automorphisms. The key ingredient is a
new local positivity property of weak solutions to the homogenuous Monge-Ampere
equation on a product domain, whose proof uses plurisubharmonic variation of
Bergman kernels.
