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 my lack of education in economics), namely in the so called Monge-Kantorovich problem. By the way, Leonid Kantorovich 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.

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. By the way, Leonid Kantorovich 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.

Monge-Amp`ere equations appear not only in geometry, but also in economics (though I cannot comment on their importance in that area), namely in the so called Monge-Kantorovich problem. By the way, Leonid Kantorovich 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.

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 my lack of education in economics), namely in the so called Monge-Kantorovich problem. By the way, Leonid Kantorovich 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.