On a lower bound of the Kobayashi metric (Q2817015)

The author presents the following characterization of convex domains in \(\mathbb C^n\) in terms of lower bounds for invariant metrics. For a domain \(D\subset\mathbb C^n\) the following conditions are equivalent: NEWLINENEWLINENEWLINE (a) \(D\) is convex; NEWLINENEWLINENEWLINENEWLINE (b) \(\gamma_D(z;X)\geq\frac1{2d_D(z;X)}\), \(z\in D\), \(X\in\mathbb C^n\), where \(\gamma_D\) stands for the Carathéodory metric and \(d_D(z;X):=\sup\big\{r>0: z+\lambda X\in D\) for all \(|\lambda|<r\big\}\); NEWLINENEWLINENEWLINENEWLINE (c) \(\liminf_{z\to a}\frac{2\varkappa_D(z;z-a)-1}{\|z-a\|}\geq0\), \(a\in\partial D\), where \(\varkappa_D\) stands for the Kobayashi metric.