Conformal inference (CI, a.k.a. conformal prediction, CP) is a statistical method which, from an iid dataset $\{Z_1, ..., Z_n\}$, provides a confidence region with a prescribed probability coverage for a new example $Z_{n+1}$. This region is defined as all the values z that are sufficiently consistent w.r.t. the dataset $\{Z_1, ..., Z_n\}$, where the said consistency is evaluated using a so-called conformity scoring function. In the regression framework, this method can be used for uncertainty quantification purposes, i.e. for obtaining a confidence region around a given predicted value. The attractiveness of CI is due to two features : (i) its coverage property in valid for any finite sample size n, and (ii) CI does not require any assumption on the distribution of the data $Z_i$. In this talk, I will introduce a new conformity score based on an empirical Mahalanobis metric, in a multivariate regression setting. The corresponding confidence regions are adaptive ellipsoidal sets. I will then discuss the properties of these confidence regions, in the large n regime in particular. When the $Z_i$ follow an elliptical distribution, I will then show that our method yields regions with a smaller volume when compared to other standard multivariate scores, and conclude with simple numerical experiments. Joint work with A. Mazoyer (IMT) and F. Gamboa (IMT).