A free boundary Monge-Amp\`ere equation and applications to complete Calabi-Yau metrics

Let P be a convex body containing the origin in its interior. We study a real Monge-Amp\`ere equation with singularities along P which is Legendre dual to a certain free boundary Monge-Amp\`ere equation. This is motivated by the existence problem for complete Calabi-Yau metrics on log Calabi-Yau pairs (X, D) with D an ample, simple normal crossings divisor. We prove the existence of solutions in C^ (P) C^1, (P), and establish the strict convexity of the free boundary. When P is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension 1 faces of P.