Geometric properties of boundary sections of solutions to the Monge--Amp\`ere equation and applications

TL;DR

This paper investigates geometric properties of boundary sections of convex solutions to the Monge-Ampère equation, deriving volume estimates, covering lemmas, and establishing a space of homogeneous type for the setting.

Contribution

It introduces new geometric properties and covering results for boundary sections of solutions to the Monge-Ampère equation, expanding the theoretical framework.

Findings

Established engulfing and separating properties of boundary sections

Proved volume estimates and a Besicovitch-type covering lemma

Showed the Monge-Ampère setting forms a space of homogeneous type

Abstract

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.