In this paper we investigate the extremal properties of the sumwhere A i are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and M varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on Γ the extremal values of the sum are obtained in terms of λ. In the case of the regular dodecahedron and icosahedron in R 3 we obtain results for which values of λ the corresponding sum is independent of the position of M on Γ. We use elementary analytic and purely geometric methods.2000 Mathematics Subject Classification. Primary: 52A40.
We consider an extremal problem in geometry. Let λ be a real number and let A, B and C be arbitrary points on the unit circle . We give a full characterization of the extremal behavior of the functionwhere M is a point on the unit circle as well. We also investigate the extremal behavior of n i=1 XP i , where the P i , for i = 1, . . . , n, are the vertices of a regular n-gon and X is a point on , concentric to the circle circumscribed around P 1 . . . P n . We use elementary analytic and purely geometric methods in the proof.
We show that the symmetrized bidisc is a C-convex domain. This provides an example of a bounded C-convex domain which cannot be exhausted by domains biholomorphic to convex domains.2000 Mathematics Subject Classification. 32F17.
Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. (2000). Primary 32F45; Secondary 32A07.
Mathematics Subject Classification
Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.
Abstract. Universal upper bounds for the Kobayashi and quasihyperbolic distances near Dini-smooth boundary points of domains in C n and R n , respectively, are obtained.
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