Geometric properties of boundary sections of solutions to the Monge–Ampère equation and applications

In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge–Ampère 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–Ampère setting forms a space of homogeneous type.