Extension of the Two-Sided Normalized probability possibility transformation to the multivariate case

This paper deals with the extension to the multivariate case of a new multivariate probability possibility transformation so called Two Sided Normalized (TSN) that have in some sense a shape similar to the cumulative distribution. TSN satisfies the consistency principle and it is the more specific that satisfies Dubois and Prade shape preservation on each side of the mode but not globally. It is applicable both for univariate and multivariate probability distributions and it is equal to the optimal distribution for symmetric distributions. The calculations are quite simple with the TSN while they can be intricate with the optimal possibility distribution in the multivariate case.