  We characterize convex isoperimetric sets in the Heisenberg group endowed
with horizontal perimeter. We first prove Sobolev regularity for a certain
class of vector fields in the plane with bounded variation, related to the
curvature equations. Then, by an approximation-reparameterization argument, we
show that the boundary of convex isoperimetric sets is foliated by geodesics of
the Carnot-Caratheodory distance.
