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.

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... tion. For any domain where E is the unit disk in C and is the Poincaré distance (also called the hyperbolic distance) on E. c D is called the Carathéodory pseudodistance of D. We also define It is well known (see [L]) that if D is a convex, bounded domain, then for any pair of points (w, z) ∈ D × D with w = z there is a c-geodesic ϕ : E → D such that ϕ(0) = w and ϕ(c * D (w, z)) = z. If D is a bounded domain, then for w ∈ D and 0 < r < 1 we define the Carathéodory ball as B c * D (w, r) := {z ∈ D : c * D (w, z) < r}. Below we shall consider the domains E(p) := {|z 1 | 2p 1 + . . . + |z n | 2p n < 1}, 1991 Mathematics Subject Classification: Primary 32H15.