A counterexample for Kobayashi completeness of balanced domains

The aim of this paper is to present an example of a bounded balanced domain of holomorphy in C" (n > 3) with continuous Minkowski function that is not Kobayashi-finitely-compact. INTRODUCTION It is known [6] that if G c C" is a bounded Reinhardt-domain of holomorphy with 0 C G then G is finitely-compact with respect to (w.r.t.) the Carath6odorydistance cG, i.e., all cG-balls are relatively compact subsets of G w.r.t. the usual topology. In a more general case, if G = Gh = {z C Cn: h(z) < 1} is a bounded balanced domain of holomorphy with continuous Minkowski function h, then G is finitely-compact w.r.t. the Bergman-distance bG [4]. On the other hand, the continuity of h is a necessary condition for G = Gh to be finitely-compact w.r.t. the Kobayashi-distance kG [5, 1]. In this paper we give an example of a bounded balanced domain of holomorphy G = Gh C C3 with continuous h that is not kG-finitely compact and therefore, not cG-finitely compact. This answers a question formulated by J. Siciak in [7]. In particular, the example shows that, in general, there is no comparison of type bG < CkG for bounded balanced domains of holomorphy with continuous Minkowski function. DEFINITIONS AND STATEMENT We repeat some of the notions that will be needed in the sequel. Definition. A domain G c Cn is called balanced' iff whenever z C G and E C, 11?