Continuity and convergence properties of extremal-interpolating disks

Let a be a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a) = infα infk j:j =k dG(αj, αk), where dG is the Gleason distance in the unit disk and the first infimum is taken over all sequences α in the unit disk which map to a by a map from the disk to the ball. The value of ρ(a) is related to whether a is an interpolating sequence with respect to analytic disks passing through it, and if a is an interpolating sequence in the ball, then ρ(a)> 0. In this work, we show that ρ(a) can be obtained as the limit of the same quantity for the truncated finite sequences, and that ρ(a) depends continuously on a when a is finite. Furthermore, we describe some of the behavior of the minimizing sequences of maps involved in the extremal problem used to define ρ