Properties of the solutions to the Monge–Ampère equation Original Research Article

We consider solutions to the equation View the MathML source when μ has a doubling property. We prove new geometric characterizations for this doubling property (by means of the so-called engulfing property) and deduce the quantitative behaviour of ϕ. Also, a constructive approach to the celebrated C1,β-estimates proved by L. Caffarelli is presented, settling one of the open questions posed by Villani (Amer. Math. Soc. 58 (2003)).