  Let (X,L) be a polarized projective complex manifold. We show, by a simple
toric one-dimensional example, that Mabuchi's K-energy functional on the
geodesically complete space of bounded positive (1,1)-forms in the first Chern
class of L, endowed with the Mabuchi metric, is not strictly convex modulo
automorphisms. However, under some further assumptions the strict convexity in
question does hold in the toric case. This leads to a uniqueness result saying
that a finite energy minimizer of the K-energy (which exists on any toric
polarized manifold (X,L) which is uniformly K-stable) is uniquely determined
modulo automorphisms under the assumption that there exists some minimizer with
strictly positive curvature current.
