John's decompositions: Selecting a large part

We extend the invertibility principle of J. Bourgain and L, Tzafriri to operators acting on arbitrary decompositions id = Sigmax(j) circle times x(j), rather than on the coordinate one. The John's decomposition brings this result to the local theory of Banach spaces. As a consequence, we get a new lemma of Dvoretzky-Rogers type, where the contact points of the unit ball with its maximal volume ellipsoid play a crucial role. We then apply these results to embeddings of l(infinity)(k) into finite dimensional spaces.