Some remarks on the indicatrix of invariant metric on convex domains

Abstract

If $\\Omega$ is a domain in $\\bold C^n$ and if (for $p\\in\\Omega$, $X\\in \\bold C^n)$ $F_\\Omega(p;X)$ denotes the infinitesimal Kobayashi metric on $\\Omega$, then the indicatrix of $\\Omega$ at $p$ is the set $I_\\Omega(p)=\\{X\\in\\bold C^n\\: F_\\Omega(p;X)<1\\}$. \n In this paper the author answers one of the questions posed by S. Kobayashi [Bull. Amer. Math. Soc. 82 (1976), no. 3, 357–416; MR0414940]. Namely, he proves that if $\\Omega$ is a (bounded or unbounded) convex domain in $\\bold C^n$, then the indicatrix of $\\Omega$ is also a convex domain in $\\bold C^n$. As an application, the author also gives an elementary proof of the classical result due to Poincaré concerning the nonequivalence of the unit ball and the polydisc in $\\bold C^n$.