A long standing Mahler's conjecture asserts that the product of the volumes of a symmetric convex body in Rn and its polar body is never less than Pn=4n/n!. Bourgain and Milman proved the lower bound cn Vn with some small positive constant c. Later, Kuperberg showed that one can take c=p/4. We shall use Hormander's ideas to give a fairly simple complex-analytic proof of the Bourgain-Milman theorem.