The condition that the
Kobayashi distance between two nearby points in a pseudo-convex domain is realized
by the Poincaré distance on a single analytic disk joining the two points is studied.
It is shown that the condition forces the Kobayashi indicatrix to be convex. Examples
of pseudo-convex domains on which this condition fails to hold are given. The
(infinitesimal) Kobayashi metric is shown to be a directional derivative of the
Kobayashi distance. It is shown that, if the condition holds near any point of a
pseudo-convex domain and if the Kobayashi metric is a complete Finsler metric of
class C2, then the Kobayashi distance between any two points in the domain can be
realized by the Poincaré distance on a single analytic disk joining the two
points.
