Abstract. Geometric lower and upper estimates are obtained for invariant metrics on C-convex domains containing no complex lines.

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.