An example of limit of Lempert Functions

The Lempert function for several poles $a_0, ..., a_N$ in a domain $\Omega$ of $\mathbb C^n$ is defined at the point $z \in \Omega$ as the infimum of $\sum^N_{j=0} \log|\zeta_j|$ over all the choices of points $\zeta_j$ in the unit disk so that one can find a holomorphic mapping from the disk to the domain $\Omega$ sending 0 to $z$. This is always larger than the pluricomplex Green function for the same set of poles, and in general different. Here we look at the asymptotic behavior of the Lempert function for three poles in the bidisk (the origin and one on each axis) as they all tend to the origin. The limit of the Lempert functions (if it exists) exhibits the following behavior: along all complex lines going through the origin, it decreases like $(3/2) \log |z|$, except along three exceptional directions, where it decreases like $2 \log |z|$. The (possible) limit of the corresponding Green functions is not known, and this gives an upper bound for it.Comment: 16 pages; references added to related work of the autho

Rigid characterizations of pseudoconvex domains

We prove that an open set $D$ in \C^n is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case.Comment: v2: Proposition 14 is improved; v3: Example 15 and the proof of Proposition 14 are change

On the zero set of the Kobayashi--Royden pseudometric of the spectral unit ball

Given $A\in\Omega_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a "generalized" tangent vector at $A$ to an entire curve in $\Omega_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $\Omega_3,$ the zero set of this metric is completely described.Comment: minor changes; to appear in Ann. Polon. Mat

"Convex" characterization of linearly convex domains

We prove that a $C^{1,1}$-smooth bounded domain $D$ in \C^n is linearly convex if and only if the convex hull of any two discs in $D$ with common center lies in $D.$Comment: to appear in Math. Scand.; v3: Appendix is adde