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The Lempert function of the symmetrized polydisc in higher dimensions is not a distance
Author(s):
Nikolai
Nikolov;
Peter
Pflug;
Wlodzimierz
Zwonek
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2921-2928.
MSC (2000):
Primary 32F45
Posted:
May 10, 2007
MathSciNet review:
2317970
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Additional information
Abstract:
We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.
References:
- 1.
- J. Agler, N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), 375-403.MR 2077158 (2005e:32022)
- 2.
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- 3.
- C. Costara, The symmetrized bidisc and Lempert's theorem, Bull. London Math. Soc. 36 (2004), 656-662. MR 2070442 (2005d:32019)
- 4.
- C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), 23-55. MR 2142182 (2006d:30054)
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- 7.
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- 9.
- L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474.MR 0660145 (84d:32036)
- 10.
- N. Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann. Polon. Math., 88 (2006), 279-283.
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Additional Information:
Nikolai
Nikolov
Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Email:
nik@math.bas.bg
Peter
Pflug
Affiliation:
Carl von Ossietzky Universität Oldenburg, Fachbereich Mathematik, Postfach 2503, D-26111 Oldenburg, Germany
Email:
pflug@mathematik.uni-oldenburg.de
Wlodzimierz
Zwonek
Affiliation:
Instytut Matematyki, Uniwersytet Jagiellonski, Reymonta 4, 30-059 Kraków, Poland
Email:
Wlodzimierz.Zwonek@im.uj.edu.pl
DOI:
10.1090/S0002-9939-07-08817-X
PII:
S 0002-9939(07)08817-X
Keywords:
Symmetrized polydisc,
Carath\'eodory distance and metric,
Kobayashi distance and metric,
Lempert function
Received by editor(s):
January 31, 2006
Received by editor(s) in revised form:
June 9, 2006
Posted:
May 10, 2007
Additional Notes:
This paper was written during the stays of the first and third named authors at Universität Oldenburg supported by grants from the DFG (January -- March 2006 and November 2005 (DFG Projekt 227/8-1/2)). They would like to thank both institutions for their support. The third author was also supported by the Research Grant No. 1 PO3A 005 28, which is financed by public means in the programme promoting science in Poland in the years 2005-2008.
The authors thank the referee for his remarks which essentially improved this paper.
Communicated by:
Mei-Chi Shaw
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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