A popular way to study the tail of a distribution is to consider its extreme quantiles. While this is a standard procedure for univariate distributions, it is harder for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile should be. In this paper, we focus on extreme geometric quantiles. Their asymptotics are established, both in direction and magnitude, under suitable moment conditions, when the norm of the associated index vector tends to one. In particular, it appears that if a random vector has a finite covariance matrix, then the magnitude of its extreme geometric quantiles grows at a fixed rate. We take advantage of these results to define an estimator of extreme geometric quantiles of such a random vector. The consistency and asymptotic normality of the estimator are established and our results are illustrated on some numerical examples
