  We introduce a notion of connected perimeter for planar sets defined as the
lower semi-continuous envelope of perimeters of approximating sets which are
measure-theoretically connected. A companion notion of simply connected
perimeter is also studied. We prove a representation formula which links the
connected perimeter, the classical perimeter, and the length of suitable
Steiner trees. We also discuss the application of this notion to the existence
of solutions to a nonlocal minimization problem.
