Abstract:Karshon constructed the first counterexample to the log-concavity conjecture for Duistermaat-Heckman measures. In this talk, we give a systematic construction of non-log-concave examples of Duistermaat-Heckman measure. In particular, we will discuss how to construct simply connected Hamiltonian manifolds which have the Hard Lefschetz property and which have a non-log-concave Duistermaat-Heckman function. On the other hand, we will explain that if there is a Hamiltonian torus action of complexity two such that all the symplectic reduced spaces taken at regular values satisfy the condition b^+=1, then its Duistermaat-Heckman function has to be log-concave. For instance, this result implies the log-concavity conjecture for Hamiltonian circle actions on six manifolds whose fixed points sets have no four dimensional peices, or have only four dimensional pieces with $b^+=1$.