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We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f\left(x\right)=in{f}_{Q\ni x}1/\left|Q\right|{\int }_Q\left|f\right|dy$, extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ℝⁿ weighted norm inequalities for the geometric maximal operator $M₀f\left(x\right)={sup}_{Q\ni x}exp(1/|Q|{\int }_{Q}log\left|f\right|dx)$, proved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal to the closely...