  We consider a cross interpolation of high-dimensional arrays in the tensor
train format. We prove that the maximum-volume choice of the interpolation sets
provides the quasioptimal interpolation accuracy, that differs from the best
possible accuracy by the factor which does not grow exponentially with
dimension. For nested interpolation sets we prove the interpolation property
and propose greedy cross interpolation algorithms. We justify the theoretical
results and test the speed and accuracy of the proposed algorithm with
convincing numerical experiments.
