Let \(D\subset\mathbb C^n\) be a domain that contains no complex lines and let \(q\in D\) be fixed. Let \(q^1\in\partial D\) be such that \(\|q^1-q\|=d_D(q)=:\tau_1(q)\), where \(d_\varOmega(z):=\text{dist}(z,\partial\varOmega)\). Let \(H_1:=q+(q^1-q)^\perp\), \(D_1:=D\cap H_1\), and let \(q^2\in\partial D_1\) be such that \(\|q^2-q\|=d_{D_1}(q)=:\tau_2(q)\). We put \(H_2:=q+\{q^1-q, q^2-q\}^\perp\), \(D_2:=D\cap H_2\) and so on. The basis \(\frac{q^j-q}{\|q^j-q\|}\), \(j=1,\dots,n\), is called the minimal basis at \(q\). The authors characterize the Carathéodory (resp. Lempert) balls \(C_D(q,r)\) (resp. \(L_D(q,r)\)) in terms of the minimal basis at \(q\). The main result of the paper is the following theorem. We assume that the above minimal basis at \(q\) coincides with the standard basis of \(\mathbb C^n\). Then:

(i)

if \(D\) is weakly linearly convex, then
\[ \mathbb D^n\Big(q,\frac1n(\tanh r)\tau(q)\Big)\subset\Big\{z:\sum_{j=1}^n\frac{|z_j-q_j|}{\tau_j(q)} <\tanh r\Big\}\subset L_D(q,r), \] where \(\mathbb D^n(q,s):=\big\{z\in\mathbb C^n: |z_j-q_j|<s_j,\;j=1,\dots,n\big\}\);

(ii)

if \(D\) is convex, then \(C_D(q,r)\subset\mathbb D^n\big(q,(e^{2r}-1)\tau(q)\big)\);

(iii)

if \(D\) is \(\mathbb C\)-convex, then \(C_D(q,r)\subset\mathbb D^n\big(q,(e^{4r}-1)\tau(q)\big)\).