Approximation to probabilities through uniform laws on convex sets (Q1397962)

Consider a probability distribution \(P\) on \(\mathbb{R}^d\), \(d\geq 1\), and denote by \({\mathcal C}\) the family of uniform probabilities supported by compact convex subsets of \(\mathbb{R}^d\) with non-empty interior. Under suitable assumptions, it is proved that there exists a best approximation to \(P\) in \({\mathcal C}\) according to the \(L_2\)-Wasserstein distance. This approximation can be considered as the best representation of \(P\) by a convex set in the minimum squares setting, which improves on other existing representations for the shape of a distribution. As a by-product, the authors derive properties related to the limit behaviour and marginals of uniform probabilities on convex subsets of \(\mathbb{R}^d\).