Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality

We provide a full quantitative version of the Gaussian isoperimetric inequality. Our estimate is independent of the dimension, sharp on the decay rate with respect to the asymmetry and with optimal dependence on the mass

Estimates on path functionals over Wasserstein spaces

In this paper we consider the class a functionals (introduced by Brancolini, Buttazzo, and Santambrogio) Gr,p(γ) defined on Lipschitz curves γ valued in the p-Wasserstein space. The problem considered is the following: given a measure μ, give conditions in order to assure the existence a curve γ such that γ(0)=μ, γ(1)=δx0, and Gr,p(γ)<+∞. To this end, new estimates on Gr,p(μ) are given and a notion of dimension of a measure (called /path dimension/) is introduced: the path dimension specifies the values of the parameters (r,p) for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions

Long-term planning

Given the probability measure ν over the given region $\Omega\subset \mathbb{R}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma\mapsto \int_\Omega \mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as $n\to\infty$, although the optimization costs in both cases have the same asymptotic orders of vanishing