  We study the discrepancy between the distribution of a vector-valued
functional of i.i.d. random elements and that of a Gaussian vector. Our main
contribution is an explicit bound on the convex distance between the two
distributions, holding in every dimension. Such a finding constitutes a
substantial extension of the one-dimensional bounds deduced in Chatterjee
(2007) and Lachi\`eze-Rey and Peccati (2017), as well as of the
multidimensional bounds for smooth test functions and indicators of rectangles
derived, respectively, in Dung (2019), and Fang and Koike (2021). Our
techniques involve the use of Stein's method, combined with a suitable
adaptation of the recursive approach inaugurated by Schulte and Yukich (2017):
this yields rates of converge that have a presumably optimal dependence on the
sample size. We develop several applications of a geometric nature, among which
is a new collection of multidimensional quantitative limit theorems for the
intrinsic volumes associated with coverage processes in Euclidean spaces.
