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We establish explicit bounds on the convex distance between the distributionof a vector of smooth functionals of a Gaussian field, and that of a normalvector with a positive definite covariance matrix. Our bounds are commensurateto the ones obtained by Nourdin, Peccati and R\'eveillac (2010) for the(smoother) 1-Wasserstein distance, and do not involve any additionallogarithmic factor. One of the main tools exploited in our work is a recursiveestimate on the convex distance recently obtained by Schulte and Yukich (2019).We illustrate our abstract results in two different situations: (i) we prove aquantitative multivariate fourth moment theorem for vectors of multipleWiener-It\^o integrals, and (ii) we characterise the rate of convergence forthe finite-dimensional distributions in the functional Breuer-Major theorem.

Multivariate normal approximation on the Wiener space: new bounds in the convex distance