For the real function \(f\) defined on \(I = [a,b]\) let \(|f |_{\alpha, \beta}\) be given by \(|f |_{\alpha, \beta} = \alpha f_+ + \beta f_-\) where \(f_\pm (x) = \max \{\pm f(x), 0\}\). When \(f \in L_1 (I)\) the authors consider the norm of \(f\) defined by \[ |f |_{1; \alpha \beta} = \int_I \bigl |f(x) \bigr |_{\alpha, \beta} dx. \] For \(f \in L_1(I)\) and \(G \subseteq L_1\) let \(P_G^{(\alpha, \beta)} (f)\) be the corresponding metric projection with respect to this norm. An element \(g \in P_G^{(\alpha, \beta)} (f)\) will be named element of \((\alpha, \beta)\)-best approximation of \(f\). The aim of this paper is to give characterizations of finite dimensional subspaces \(G\) of \(C(I)\) or \(C^1 (I)\) (containing or not the constants) such that every \(f \in L_1 (I)\) (or \(f \in C (I)\); \(f \in C^1 (I))\) to have a unique element of \((\alpha, \beta) \)-best approximation or one-sided best approximation, in \(G\). In particular, known results in the literature are obtained.