Geometric properties of boundary sections of solutions to the Monge-Ampere equation and applications

In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Ampere equation: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of Besicovitch type, a covering theorem and a strong type p-p estimate for the maximal function corresponding to boundary sections. Moreover, we show that the Monge-Ampere setting forms a space of homogeneous type. Published by Elsevier Inc.