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In the present work, we show how the generalized Cram\'er-Rao inequality forthe estimation of a parameter, presented in a recent paper, can be extended tothe mutidimensional case with general norms on $\mathbb{R}^{n}$, and to a widercontext. As a particular case, we obtain a new multidimensional Cram\'er-Raoinequality which is saturated by generalized $q$-Gaussian distributions. Wealso give another related Cram\'er-Rao inequality, for a general norm, which issaturated as well by these distributions. Finally, we derive uncertaintyrelations from these Cram\'er-Rao inequalities. These uncertainty relationsinvolve moments computed with respect to escort distributions, and we show thatsome of these relations are saturated by generalized $q$-Gaussiandistributions. These results introduce extended versions of Fisher information,new Cram\'er-Rao inequalities, and new characterizations of generalized$q$-Gaussian distributions which are important in several areas of physics andmathematics.

On multidimensional generalized Cram\'er-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions