  We characterise, in terms of their transition laws, the class of
one-dimensional L\'evy processes whose graph has a continuously differentiable
(planar) convex hull. We show that this phenomenon is exhibited by a broad
class of infinite variation L\'evy processes and depends subtly on the
behaviour of the L\'evy measure at zero. We introduce a class of strongly
eroded L\'evy processes, whose Dini derivatives vanish at every local minimum
of the trajectory for all perturbations with a linear drift, and prove that
these are precisely the processes with smooth convex hulls. We study how the
smoothness of the convex hull can break and construct examples exhibiting a
variety of smooth/non-smooth behaviours. Finally, we conjecture that an
infinite variation L\'evy process is either strongly eroded or abrupt, a claim
implied by Vigon's point-hitting conjecture. In the finite variation case, we
characterise the points of smoothness of the hull in terms of the L\'evy
measure.
