On different extremal bases for \(\mathbb C\)-convex domains (Q2845470)

Let \(D\subset\mathbb C^n\) be a domain, \(q\in D\), \(d_D(q;a):=\sup\{r>0: q+r\mathbb Da\subset D\}\), \(a\in\mathbb C^n\) (\(\mathbb D\) stands for the unit disc). We assume that \(\partial D\) is so regular that \(d_D(q;\cdot)\) is continuous. Take a \(p_1\in\partial D\) such that \(m_1:=\|p_1-q\|=\) the Euclidean distance of \(q\) to \(\partial D\). Let \(a_1:=(p_1-q)/m_1\), \(H_1:=q+a_1^\perp\). Put \(D_2:=D\cap H_1\) and let \(a_2\in\{a_1\}^\perp\), \(\|a_2\|=1\), be such that NEWLINENEWLINE\[NEWLINEm_2:=\sup\big\{d_D(q;a): a\in a_1^\perp,\;\|a\|=1\big\}=d_D(q,a_2).NEWLINE\]NEWLINE NEWLINESet \(p_2:=q+m_2a_2\in\partial D_2\). Put \(H_2:=q+\{a_1,a_2\}^\perp\), \(D_3:=D\cap H_2\) and repeat the procedure. Finally, we get an orthonormal basis \(a_1,\dots,a_n\) (called the maximal basis of \(D\) at \(q\)), numbers \(m_1\geq\dots\geq m_n\), and boundary points \(p_j=q+m_ja_j\), \(j=1,\dots,n\).NEWLINENEWLINELet now \(D\) be convex and \(\mathcal C^\infty\)-smooth in a neighborhood of \(p_1\) and \(q\) belongs to the inner normal at \(p_1\), sufficiently near to \(p_1\). Let \(r\) be a defining function in a neighborhood of \(p_1\) and let \(z=q+\sum_{j=1}w_ja_j\) be a coordinate system given by the maximal basis. It was claimed that then (*) \(\frac{\partial r}{\partial w_j}(p_k)=0\) for \(j=k+1,\dots,n\). The authors construct an example of a smooth strictly convex domain in \(\mathbb C^3\) such that (*) fails to hold. Nevertheless, they prove that the estimates for the Bergman kernel and invariant metrics, the proofs of which were based on the incorrect property (*), remain true.