  We establish a novel characterisation of the law of the convex minorant of
any L\'evy process. Our self-contained elementary proof is based on the
analysis of piecewise linear convex functions and requires only very basic
properties of L\'evy processes. Our main result provides a new simple and
self-contained approach to the fluctuation theory of L\'evy processes,
circumventing local time and excursion theory. Easy corollaries include
classical theorems, such as Rogozin's regularity criterion, Spitzer's
identities and the Wiener-Hopf factorisation, as well as a novel factorisation
identity.
