Carathéodory balls in convex complex ellipsoids

Włodzimierz Zwonek

Annales Polonici Mathematici, Tome 63 (1996), p. 183-194 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 1996-01-01

Zbl 0864.32017

EUDML-ID : urn:eudml:doc:269975

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     author = {W\l odzimierz Zwonek},
     title = {Carath\'eodory balls in convex complex ellipsoids},
     journal = {Annales Polonici Mathematici},
     volume = {63},
     year = {1996},
     pages = {183-194},
     zbl = {0864.32017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv64z2p183bwm}
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Włodzimierz Zwonek. Carathéodory balls in convex complex ellipsoids. Annales Polonici Mathematici, Tome 63 (1996) pp. 183-194. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv64z2p183bwm/

Bibliographie

[000] [JP] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter, 1993. | Zbl 0789.32001

[001] [JPZ] M. Jarnicki, P. Pflug and R. Zeinstra, Geodesics for convex complex ellipsoids, Ann. Scuola Norm. Sup. Pisa 20 (1993), 535-543. | Zbl 0812.32010

[002] [L] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-479.

[003] [R] W. Rudin, Function Theory in the Unit Ball of $ℂ^{n}$ , Grundlehren Math. Wiss. 241, Springer, 1980.

[004] [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.

[005] [Sc-Sr] B. Schwarz and U. Srebro, Carathéodory balls and norm balls in $H_{p, n} = z \in ℂ^{n} : ‖ z ‖_{p} < 1$ , preprint.

[006] [Sr] U. Srebro, Carathéodory balls and norm balls in $H = z \in ℂ^{n} : ‖ z ‖ ₁ < 1$ , Israel J. Math. 89 (1995), 61-70.

[007] [Z] W. Zwonek, Carathéodory balls and norm balls of the domains $H_{n} = z \in ℂ^{n} : | z ₁ | + . . . + | z_{n} | < 1$ , Israel J. Math. 89 (1995), 71-76. | Zbl 0824.32007