Geometric properties of boundary sections of solutions to the   Monge--Amp\`ere equation and applications
  In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Amp\`ere equations: 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-Amp\`ere setting forms a space of homogeneous type. 
Nam Q. Le, Truyen Nguyen
math.AP