We build a sharp approximation of the whole distribution of the sum of iid\nheavy-tailed random vectors, combining mean and extreme behaviors. It extends\nthe so-called 'normex' approach from a univariate to a multivariate framework.\nWe propose two possible multi-normex distributions, named $d$-Normex and\nMRV-Normex. Both rely on the Gaussian distribution for describing the mean\nbehavior, via the CLT, while the difference between the two versions comes from\nusing the exact distribution or the EV theorem for the maximum. The main\ntheorems provide the rate of convergence for each version of the multi-normex\ndistributions towards the distribution of the sum, assuming second order\nregular variation property for the norm of the parent random vector when\nconsidering the MRV-normex case. Numerical illustrations and comparisons are\nproposed with various dependence structures on the parent random vector, using\nQQ-plots based on geometrical quantiles.\n