  A nonlinear analogue of the Rademacher type of a Banach space was introduced
in classical work of Enflo. The key feature of Enflo type is that its
definition uses only the metric structure of the Banach space, while the
definition of Rademacher type relies on its linear structure. We prove that
Rademacher type and Enflo type coincide, settling a long-standing open problem
in Banach space theory. The proof is based on a novel dimension-free analogue
of Pisier's inequality on the discrete cube.
