  We derive tight non-asymptotic bounds for the Kolmogorov distance between the
probabilities of two Gaussian elements to hit a ball in a Hilbert space. The
key property of these bounds is that they are dimension-free and depend on the
nuclear (Schatten-one) norm of the difference between the covariance operators
of the elements and on the norm of the mean shift. The obtained bounds
significantly improve the bound based on Pinsker's inequality via the
Kullback-Leibler divergence. We also establish an anti-concentration bound for
a squared norm of a non-centered Gaussian element in Hilbert space. The paper
presents a number of examples motivating our results and applications of the
obtained bounds to statistical inference and to high-dimensional CLT.
