# Carathéodory balls in convex complex ellipsoids

Annales Polonici Mathematici (1996)

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

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topWłodzimierz 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

top- [JP] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter, 1993. Zbl0789.32001
- [JPZ] M. Jarnicki, P. Pflug and R. Zeinstra, Geodesics for convex complex ellipsoids, Ann. Scuola Norm. Sup. Pisa 20 (1993), 535-543. Zbl0812.32010
- [L] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-479.
- [R] W. Rudin, Function Theory in the Unit Ball of ${\u2102}^{n}$, Grundlehren Math. Wiss. 241, Springer, 1980.
- [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.
- [Sc-Sr] B. Schwarz and U. Srebro, Carathéodory balls and norm balls in ${H}_{p,n}=z\in {\u2102}^{n}:\Vert z{\Vert}_{p}<1$, preprint.
- [Sr] U. Srebro, Carathéodory balls and norm balls in $H=z\in {\u2102}^{n}:\Vert z\Vert \u2081<1$, Israel J. Math. 89 (1995), 61-70.
- [Z] W. Zwonek, Carathéodory balls and norm balls of the domains ${H}_{n}=z\in {\u2102}^{n}:\left|z\u2081\right|+...+|{z}_{n}|<1$, Israel J. Math. 89 (1995), 71-76. Zbl0824.32007

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