Geometric properties of boundary sections of solutions to the Monge-Ampère equation and applications (Q1932410)

The authors establish several geometric properties of boundary sections of convex solutions to the Monge-Ampére equation \[ \text{det}\, D^2\varphi =g \quad \text{ in } \quad \Omega, \] with right hand side bounded away from \(0\) and \(\infty\) and with smooth boundary data: the engulfing and separating properties and volume estimates. As applications, the authors 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, it is shown that the Monge-Ampére setting forms a space of homogeneous type.