Non-linear factorization of linear operators

We show, in particular, that a linear operator between finite-dimensional normed spaces, which factors through a third Banach space Z via Lipschitz maps, factors linearly through the identity from L-infinity([0, 1], Z) to L-1([0, 1], Z) (and thus, in particular, through each L-p(Z), for 1 < p < infinity) with the same factorization constant. It follows that, for each 1 < p < infinity, the class of L-p spaces is closed under uniform (and even coarse) equivalences. The case p = 1 is new and solves a problem raised by Heinrich and Mankiewicz in 1982. The proof is based on a simple local-global linearization idea.