  Let $P$ be a convex body containing the origin in its interior. We study a
real Monge-Amp\`ere equation with singularities along $\del P$ 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 $C^{\infty}(P)\cap
C^{1,\alpha}(\overline{P})$, 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 $\del P$.
