### The Monge problem on non-compact manifolds

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In this note we provide a new geometric lower bound on the so-called Grad’s number of a domain $\xd8$ in terms of how far $\xd8$ is from being axisymmetric. Such an estimate is important in the study of the trend to equilibrium for the Boltzmann equation for dilute gases.

In this note we provide a new geometric lower bound on the so-called Grad's number of a domain in terms of how far is from being axisymmetric. Such an estimate is important in the study of the trend to equilibrium for the Boltzmann equation for dilute gases.

We prove that any Kantorovich potential for the cost function = /2 on a Riemannian manifold (, ) is locally semiconvex in the “region of interest”, without any compactness assumption on , nor any assumption on its curvature. Such a region of interest is of full -measure as soon as the starting measure does not charge – 1-dimensional rectifiable sets.

We study points of density $1/2$ of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density $1/2$ is formulated in terms of the pointwise behaviour of the Ornstein-Uhlembeck semigroup.

Given a measurable set $A\subset {\mathbb{R}}^{n}$ of positive measure, it is not difficult to show that $|A+A|=\left|2A\right|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$.

The irrigation problem is the problem of finding an efficient way to transport a measure μ onto a measure μ. By efficient, we mean that a structure that achieves the transport (which, following [Bernot, Caselles and Morel, (2005) 417–451], we call traffic plan) is better if it carries the mass in a grouped way rather than in a separate way. This is formalized by considering costs functionals that favorize this property. The aim of this paper is to introduce a dynamical cost functional...

This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps....

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