We provide three characterizations of the minimal martingale measure associated to a given d-dimensional semimartingale X. In each case, is shown to be the unique solution of an optimization problem where one minimizes a certain functional over a suitable class of signed local martingale measures for X. Furthermore, we extend a result of Ansel and Stricker on the Fllmer-Schweizer decomposition to the case where X is continuous, but multidimensional.