The authors use the following notation: \({I_\epsilon:=\{x\in{\mathbb R}^k:\;-\epsilon\leq x_i\leq\epsilon\} }\), \(B_\epsilon\) is the closed ball of radius \(\epsilon\) centered at \(0\), \(\Pi^r\) denotes the set of polynomials of degree at most \(r\). A function \(w\) is a weight function if it is positive a.e. on \(I_1\) and \({w\in L^1(I_1)}\). Let \({\phi:[0,\infty)\to [0,\infty)}\) be a convex function, \({\phi(0)=0}\), and \(\phi\) satisfies some other conditions. For a measurable function \(g\), a measurable set \(A\), and a weight \(w\) the functional \({F_{w,A}}\) is defined by \[ F_{w,A}(g):=\left(\int_A w(t)dt\right)^{-1}\int_A\phi(| g(t)| )w(t)dt. \] The Luxemburg norm of \(g\) is defined by \[ \| g\| _{w,A}:=\inf\left\{\lambda>0:\;F_{w,A}(g\lambda^{-1})\leq 1\right\}. \] Let \({W(\epsilon):=\int_{B_\epsilon}w(t)dt}\) and \({w_\epsilon(t):=\epsilon^k W(\epsilon)^{-1} w(\epsilon t)}\) for \({\epsilon>0}\). It is assumed that \(w(t)\) is a radial weight function and there exist two real numbers \(\lambda\) and \(\beta\) such that \(\lambda>0\), \({\beta+k>0}\) and \({W(\epsilon)=\lambda \epsilon^{\beta+k}(1+o(1))}\). A new weight is defined by \({\tilde w(t):=w_k^{-1}(\beta+k)| t| ^\beta}\), where \(w_k\) is the surface area of \(B_1\). In Section 2 the authors prove that under some conditions on \(f\) and \(\phi\) the Luxemburg norms \({\| \cdot\| _{w_\epsilon,B_1}}\) are uniformly equivalent to the Luxemburg norm \({\| \cdot\| _{\tilde w,B_1}}\) on the linear space \[ E_f:=\{fQ-P:\;(P,Q)\in\Pi^n\times\Pi^m\}. \] In Section 3 the authors consider the problem of minimizing \({\| fQ-P\| _{w,B_\epsilon}}\) when \({(P,Q)}\) runs a certain subset of \({\Pi^n\times\Pi^m}\). The concept of Padé approximant of a function \(f\) with respect to the Luxemburg norm is introduced. It is proved that under some conditions the net of rational functions \({P_\epsilon/Q_\epsilon}\) converges in the Luxemburg norm to Padé approximant as \({\epsilon\to 0}\). Here the pair \({(P_\epsilon,Q_\epsilon)}\) is a solution of the minimization problem. In Section 4 the authors characterize the limit of the error function for the best polynomial approximant with respect to the Luxemburg norm. They also give a new condition on a weight in order to obtain inequalities of the type \({\| P\| _\infty\leq K\epsilon^{-n}\| P\| _{w,I_\epsilon} }\), for all \({P\in\Pi^n}\), where \(K\) is a constant independent of \(P\) and \(\epsilon\). These inequalities play an important role in problems of weighted best local approximation.