  In this paper we consider Monge-Amp\`ere equations on compact Hessian
manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex
domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by
the action of an affine group. In the case where the affine group action is
volume-preserving, i.e., when the manifold is special, the solvability of the
corresponding Monge-Amp\`ere equation was established using the continuity
method by Cheng and Yau. In the general case we set up a variational framework
involving certain dual manifolds and a generalization of the classical Legendre
transform. We give existence and uniqueness results, elaborate on connections
to optimal transport and quasi-periodic tilings of convex domains.
