  In this paper, we consider the estimation of a change-point for possibly
high-dimensional data in a Gaussian model, using a k-means method. We prove
that, up to a logarithmic term, this change-point estimator has a minimax rate
of convergence. Then, considering the case of sparse data, with a Sobolev
regularity, we propose a smoothing procedure based on Lepski's method and show
that the resulting estimator attains the optimal rate of convergence. Our
results are illustrated by some simulations. As the theoretical statement
relying on Lepski's method depends on some unknown constant, practical
strategies are suggested to perform an optimal smoothing.
