Intrinsic metrics in complete circular domains (Q584468)

Let \(D\) be a complete circular domain in a locally convex separate topological vector space. Let \(k_ D\) and \(c_ D\) be respectively the Kobayashi and Carathéodory pseudodistance and let \(\kappa_ D\) and \(\gamma_ D\) be the corresponding infinitesimal pseudometrics; denote also by \(k^ 1_ D\) the ``one-disc'' Kobayashi pseudodistance, that is \[ k^ 1_ D(z,w)=\inf \{k^ 1_{\Delta}(\xi,\eta)| \quad \exists f\in Hol(\Delta,D)s.t.f(\xi)=z,\quad f(\eta)=w\},\quad \Delta =\{f\in {\mathbb{C}}| \quad | z| <1\}, \] and let \(\mu\) and \({\hat \mu}\) be respectively the Minkowski functional associated to D and \(\hat D,\) the convex hull of \(D\). The main results of the paper are the following: (1) \(\lim_{t\to 0}(1/| t|)k_ D(tX,tY)=\lim_{t\to 0}(1/| t|)c_ D(tX,tY)=\gamma_ D(0,X-Y)\) \(\forall X,Y\in V\) (2) given \(X\in V\), if \(\mu\) (X)\(\neq 0\) then \(\lim_{t\to 0} k_ D(0,tX)/k^ 1_ D(0,tX)= {\hat \mu}(X)/\mu (X)\) (3) given \(z\in D\) one has \(k_ D(0,z)=k^ 1_ D(0,z)\) iff \(\mu(z)={\hat\mu}(z)\), and in this case one has \(k_ D(0,z)=c_ D(0,z)\).