Lempert Theorem for stronly linearly convex domains with smooth boundaries

TL;DR

This paper provides a detailed, accessible proof of the Lempert Theorem for non-planar strongly linearly convex domains with smooth boundaries, clarifying previous sketchy proofs and exploring implications for complex domain exhaustion.

Contribution

It offers an extended, clarified proof of the Lempert Theorem for smooth strongly linearly convex domains, addressing gaps in earlier presentations and implications for domain exhaustion.

Findings

Extended proof of Lempert Theorem for smooth strongly linearly convex domains

Demonstrates the exhaustion of the symmetrized bidisc by such domains

Shows the non-exhaustibility of the bidisc by biholomorphic convex domains

Abstract

The aim of this paper is to present a detailed and slightly modified version of the proof of the Lempert Theorem in the case of non-planar stronlgy linearly convex domains with C^2 smooth boundaries. The original Lempert's proof is presented only in proceedings of a conference 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. Additional motivation for presenting the proof is the fact shown recently, that the so-called symmetrized bidisc may be exhausted by stronlgy linearly convex domains. On the other hand it cannot be exhausted by domains biholomorphic to convex ones. Therefore, the equality of the Lempert function and the Carath\'eodory distance for strongly linearly convex domains does not follow directly from the convex case.