Monge-Amp\`ere equations appear not only in geometry, but also in
**economics** (though I cannot comment on their importance in that
area due to lack of my education in economics), namely in the so called [Monge-Kantorovich problem][1]. By the way,
[Leonid Kantorovich][2] was a mathematician and economist who received a
Nobel prize in economics.

The problem is as follows. Let $\mu_1,\mu_2$ be two probability
measures in $\mathbb{R}^n$. We are looking for a measurable map
$f\colon \mathbb{R}^n\rightarrow\mathbb{R}^n$ such that
$f_*(\mu_1)=\mu_2$ (where $f_*$ is the usual push-forward on
measures), and $f$ minimizes certain cost functional.

Brenier has shown existence of such a map (called now the Brenier
map) under appropriate conditions on the measures and the cost
functional; he reduced the problem to solvability of certain
Monge-Amp\`ere equation. Other people proved some regularity of the
solution.

The Brenier map was applied further by F. Barthe  in a completely
different area: **to prove  a new functional inequality** called the
inverse Brascamp-Lieb inequality (see "On a reverse form of the
Brascamp-Lieb inequality", Invent. Math. 134 (1998), no. 2,
335–-361). He also obtained with his method a new proof of the known
Brascamp-Lieb inequality. Moreover in the same paper, Barthe deduced
from his functional inequality a new isoperimetric property of
simplex and parallelotop: simplex is the ONLY convex body with
minimal volume ratio, while parallelotope is the ONLY centrally
symmetric convex body with minimal volume ratio. (Previously  K.
Ball has shown these minimality properties of simplex and
parallelotop without proving the uniqueness, using a different
technique.) Remind that volume ratio of a convex body is, by
definition, the ratio of its volume to the volume of ellipsoid of
maximal volume contained in it.

Later on other authors applied the Brenier map to obtain sharp
constants in some other functional inequalities.


  [1]: http://en.wikipedia.org/wiki/Transportation_theory
  [2]: http://en.wikipedia.org/wiki/Leonid_Kantorovich