  Let M be a compact, connected symplectic 2n-dimensional manifold on which
an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the
assumption that there is an effective complementary 2-torus acting on M with
symplectic orbits, we show that the Duistermaat-Heckman measure of the T-action
is log-concave. This verifies the logarithmic concavity conjecture for a class
of inequivalent T-actions. Then we use this conjecture to prove the following:
if there is an effective symplectic action of an (n-2)-dimensional torus T on a
compact, connected symplectic 2n-dimensional manifold that admits an effective
complementary symplectic action of a 2-torus with symplectic orbits, then the
existence of T-fixed points implies that the T-action is Hamiltonian. As a
consequence of this, we give new proofs of a classical theorem by McDuff about
S^1-actions, and some of its recent extensions.
