Given two measured spaces $(X,\mu )$ and $(Y,\nu )$, and a third space $Z$, given two functions $u(x,z)$ and $v(x,z)$, we study the problem of finding two maps $s:X\to Z$ and $t:Y\to Z$ such that the images $s\left(\mu \right)$ and $t\left(\nu \right)$ coincide, and the integral ${\int }_{X}u(x,s\left(x\right))d\mu -{\int }_{Y}v(y,t\left(y\right))d\nu$ is maximal. We give condition on $u$ and $v$ for which there is a unique solution.

Given two measured spaces $(X,\mu )$ and $(Y,\nu )$, and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps $s:X\to Z$ and $t:Y\to Z$ such that the images $s\left(\mu \right)$ and $t\left(\nu \right)$ coincide, and the integral ${\int }_{X}u(x,s\left(x\right))d\mu -{\int }_{Y}v(y,t\left(y\right))d\nu$ is maximal. We give condition on u and v for which there is a unique solution.

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of a metric measure space...