  Given a matrix U, using a deterministic method, we extract a "large"
submatrix of U'(whose columns are obtained by normalizing those of U) and
estimate its smallest and largest singular value. We apply this result to the
study of contact points of the unit ball with its maximal volume ellipsoid. We
consider also the paving problem and give a deterministic algorithm to
partition a matrix into almost isometric blocks recovering previous results of
Bourgain-Tzafriri and Tropp. Finally, we partially answer a question raised by
Naor about finding an algorithm in the spirit of Batson-Spielman-Srivastava's
work to extract a "large" square submatrix of "small" norm.
