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

Abstract

In this paper we establish several geometric properties of the cross sections of generalized solutions ϕ \phi to the Monge-Ampère equation det D 2 ϕ = μ \det D^{2}\phi = \mu , when the measure μ \mu satisfies a doubling property. A main result is a characterization of the doubling measures μ \mu in terms of a geometric property of the cross sections of ϕ \phi . This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations.