Let \(f\) be a convex symmetric radial function on \(\mathbb{R}^n\). Thus \(f\) determines a norm on \(\mathbb{R}^n\). For \(a\in \mathbb{R}^n\) and \(\lambda>0\), define the set \(B_f^\lambda(a)= \{y\in \mathbb{R}^n\mid f(y-a)\leq \lambda\}\). For \(\alpha= (\alpha_1,\dots, \alpha_n)\in \mathbb{R}^n\), a point \(\tau= (p,q)= (p,a_1,\dots, a_n)\in \mathbb{Z}^{n+1}\) is called an \(f\)-best approximation to \(\alpha\), if \(p\geq 1\) and \(f(\alpha q-b)> f(\alpha p-a)\) for all \((q,b)= (q, b_1,\dots, b_n)\in \mathbb{Z}^{n+1}\) with \((q,b)\neq (p,a)\) and \(1\leq q\leq p\). All \(f\)-best approximations to \(\alpha\) can be arranged in sequences \[ \tau_\nu= (p_\nu, a_\nu)= (p_\nu, a_{1\nu},\dots, a_{n\nu})\in \mathbb{Z}^{n+1}, \qquad 1\leq p_1 f(\alpha p_2-a_2)>\cdots> f(\alpha p_\nu- a_\nu)>\cdots\;. \] We set \(G_\nu= (G_{1\nu},\dots, G_{n\nu})\), where \(G_{j\nu}= (\alpha_j p_\nu- a_{j\nu})/ f(\alpha p_\nu- a_\nu)\). The author proves the following results: (i) \(G_{k+1}\not\in \operatorname {int} B_f^1 (G_k)\) for all \(k\in \mathbb{N}\). (ii) If \(f\) is a strictly convex norm, then there exists \(\delta= \delta(f)> 0\) such that for every \(\alpha \not\in \mathbb{Q}^n\) there are infinitely many numbers \(k\) satisfying \(G_{k+1} \not\in B_f^{1+ \delta} (G_k)\). (iii) If \(f\) is a strictly convex norm and \(\alpha\) is badly approximable, then there exist constants \(w= w(\alpha,f)\in \mathbb{N}\) and \(\delta= \delta (\alpha,f)> 0\) with the following property: for every \(k\geq 1\), there is an integer \(j\in [k, k+w]\) satisfying \(G_{j+1} \not\in B_f^{1+\delta} (G_j)\). Furthermore, the author also considers a theorem on signatures and asymptotic directions for the \(f\)-best approximations.