Abstract. In 1984 L. Lempert showed that the Lempert function and the Carathéodory distance coincide on non-planar bounded strongly linearly convex domains with real analytic boundaries. Following this paper, we present a slightly modified and more detailed version of the proof. Moreover, the Lempert Theorem is proved for non-planar bounded C 2 -smooth strongly linearly convex domains.The aim of this paper is to present a detailed version of the proof of the Lempert Theorem in the case of non-planar bounded strongly linearly convex domains with smooth boundaries. The original Lempert's proof is presented only in proceedings of a conference (see [6]) with a very limited access and at some places it was quite sketchy. We were encouraged by some colleagues to prepare an extended version of the proof in which all doubts could be removed and some of details of the proofs could be simplified. We hope to have done it below. Certainly, the idea of the proof belongs entirely to Lempert. The main differences, we would like to draw attention to, are• results are obtained in C 2 -smooth case; • the notion of stationary mappings and E-mappings is separated;• a geometry of domains is investigated only in neighborhoods of boundaries of stationary mappings (viewed as boundaries of analytic discs) -this allows us to obtain localization properties for stationary mappings; • boundary properties of strongly convex domains are expressed in terms of the squares of their Minkowski functionals.Additional motivation for presenting the proof is the fact, showed recently in [7], that the so-called symmetrized bidisc may be exhausted by strongly linearly convex domains. On the other hand it cannot be exhausted by domains biholomorphic to convex ones ([1]). Therefore, the equality of the Lempert function and the Carathéodory distance for strongly linearly convex domains does not follow directly from [5].

No abstract

We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain D increases asymptotically at most like − log d D + C. Moreover, for boundary points from intD the growth does not exceed 1 2 log(− log d D ) + C. The lower estimate by − 1 2 log d D + C is obtained under additional assumptions of C 1 -smoothness of a domain and a non-tangential convergence.2010 Mathematics Subject Classification. Primary: 32F45. Secondary: 32A07.

Abstract. In the paper we study the geometry of semitube domains in C 2 . In particular, we extend the result of Burgués and Dwilewicz for semitube domains dropping out the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains obtaining among others a relation between pseudoconvexity of a semitube domain and the number of components of its vertical slices.Finally, we present an example showing that there is a non-convex domain in C n such that its image under arbitrary isometry is pseudoconvex.