  Mathematicians have been proposing for sometimes that Monge-Amp\`ere
equation, a nonlinear generalization of the Poisson equation, where trace of
the Hessian is replaced by its determinant, provides an alternative
non-relativistic description of gravity. Monge-Amp\`ere equation is affine
invariant, has rich geometric properties, connects to optimal transport theory,
and remains bounded at short distances. Monge-Amp\`ere gravity, that uses a
slightly different form of the Monge-Amp\`ere equation, naturally emerges
through the application of large-deviation principle to a Brownian system of
indistinguishable and independent particles. In this work we provide a physical
formulation of this mathematical model, study its theoretical viability and
confront it with observations. We show that Monge-Amp\`ere gravity cannot
replace the Newtonian gravity as it does not withstand the solar-system test.
We then show that Monge-Amp\`ere gravity can describe a scalar field, often
evoked in modified theories of gravity such as Galileons. We show that
Monge-Amp\`ere gravity, as a nonlinear model of a new scalar field, is screened
at short distances, and behaves differently from Newtonian gravity above
galactic scales but approaches it asymptotically. Finally, we write a
relativistic Lagrangian for Monge-Amp\`ere gravity in flat space time, which is
the field equation of a sum of the Lagrangians of all Galileons. We also show
how the Monge-Amp\`ere equation can be obtained from the fully covariant
Lagrangian of quartic Galileon in the static limit. The connection between
optimal transport theory and modified theories of gravity with second-order
field equations, unravelled here, remains a promising domain to further
explore.
