Geometric properties of the sections of solutions to the Monge-Ampere equation

In this paper we establish several geometric properties of the cross sectio ns of generalized solutions phi to the Monge-Ampere equation det D-2 phi = mu, when the measure mu satisfies a doubling property. A main result is a c haracterization of the doubling measures mu in terms of a geometric propert y of the cross sections of phi. This is used to obtain estimates of the sha pe and invariance properties of the cross sections that are valid under app ropriate normalizations.