Carathéodory balls in convex complex ellipsoids

Włodzimierz Zwonek

Annales Polonici Mathematici (1996)

  • Volume: 64, Issue: 2, page 183-194
  • ISSN: 0066-2216

Abstract

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We consider the structure of Carathéodory balls in convex complex ellipsoids belonging to few domains for which explicit formulas for complex geodesics are known. We prove that in most cases the only Carathéodory balls which are simultaneously ellipsoids "similar" to the considered ellipsoid (even in some wider sense) are the ones with center at 0. Nevertheless, we get a surprising result that there are ellipsoids having Carathéodory balls with center not at 0 which are also ellipsoids.

How to cite

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Włodzimierz Zwonek. "Carathéodory balls in convex complex ellipsoids." Annales Polonici Mathematici 64.2 (1996): 183-194. <http://eudml.org/doc/269975>.

@article{WłodzimierzZwonek1996,
abstract = {We consider the structure of Carathéodory balls in convex complex ellipsoids belonging to few domains for which explicit formulas for complex geodesics are known. We prove that in most cases the only Carathéodory balls which are simultaneously ellipsoids "similar" to the considered ellipsoid (even in some wider sense) are the ones with center at 0. Nevertheless, we get a surprising result that there are ellipsoids having Carathéodory balls with center not at 0 which are also ellipsoids.},
author = {Włodzimierz Zwonek},
journal = {Annales Polonici Mathematici},
keywords = {Carathéodory ball; c-geodesic; convex complex ellipsoid; Carathéodory balls; convex complex ellipsoids},
language = {eng},
number = {2},
pages = {183-194},
title = {Carathéodory balls in convex complex ellipsoids},
url = {http://eudml.org/doc/269975},
volume = {64},
year = {1996},
}

TY - JOUR
AU - Włodzimierz Zwonek
TI - Carathéodory balls in convex complex ellipsoids
JO - Annales Polonici Mathematici
PY - 1996
VL - 64
IS - 2
SP - 183
EP - 194
AB - We consider the structure of Carathéodory balls in convex complex ellipsoids belonging to few domains for which explicit formulas for complex geodesics are known. We prove that in most cases the only Carathéodory balls which are simultaneously ellipsoids "similar" to the considered ellipsoid (even in some wider sense) are the ones with center at 0. Nevertheless, we get a surprising result that there are ellipsoids having Carathéodory balls with center not at 0 which are also ellipsoids.
LA - eng
KW - Carathéodory ball; c-geodesic; convex complex ellipsoid; Carathéodory balls; convex complex ellipsoids
UR - http://eudml.org/doc/269975
ER -

References

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  1. [JP] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter, 1993. Zbl0789.32001
  2. [JPZ] M. Jarnicki, P. Pflug and R. Zeinstra, Geodesics for convex complex ellipsoids, Ann. Scuola Norm. Sup. Pisa 20 (1993), 535-543. Zbl0812.32010
  3. [L] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-479. 
  4. [R] W. Rudin, Function Theory in the Unit Ball of ℂ n , Grundlehren Math. Wiss. 241, Springer, 1980. 
  5. [Sc] B. Schwarz, Carathéodory balls and norm balls of the domain H = {(z₁,z₂) ∈ ℂ²: |z₁| + |z₂| < 1}, Israel J. Math. 84 (1993), 119-128. 
  6. [Sc-Sr] B. Schwarz and U. Srebro, Carathéodory balls and norm balls in H p , n = z ∈ ℂ n : ‖ z ‖ p < 1 , preprint. 
  7. [Sr] U. Srebro, Carathéodory balls and norm balls in H = z ∈ ℂ n : ‖ z ‖ ₁ < 1 , Israel J. Math. 89 (1995), 61-70. 
  8. [Z] W. Zwonek, Carathéodory balls and norm balls of the domains H n = z ∈ ℂ n : | z ₁ | + . . . + | z n | < 1 , Israel J. Math. 89 (1995), 71-76. Zbl0824.32007

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