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Extension Property and Universal Sets

Part of: Genetics and population dynamics Analytic continuation Geometric convexity

Published online by Cambridge University Press:  24 February 2020

Łukasz Kosiński  and
Włodzimierz Zwonek
Łukasz Kosiński
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland e-mail: lukasz.kosinski@uj.edu.plwlodzimierz.zwonek@uj.edu.pl
Włodzimierz Zwonek
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland e-mail: lukasz.kosinski@uj.edu.plwlodzimierz.zwonek@uj.edu.pl
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Abstract

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Motivated by works on extension sets in standard domains, we introduce a notion of the Carathéodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a class of two-dimensional submanifolds in the tridisc that not only turns out to be Carathéodory but also provides examples of two-dimensional domains for which the celebrated Lempert Theorem holds. Additionally, a recently introduced notion of universal sets for the Carathéodory extremal problem is studied and new results on domains admitting (no) finite universal sets are given.

Keywords

extension setCarathéodory setLempert theoremuniversal set for the Carathéodory extremal problem

MSC classification

Primary: 32D15: Continuation of analytic objects
Secondary: 32F45: Invariant metrics and pseudodistances
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 717 - 736
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

The first author is partially supported by NCN grant SONATA BIS no. 2017/26/E/ST1/00723. The second author is partially supported by the OPUS grant no. 2015/17/B/ST1/00996 of the National Science Centre, Poland.

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