  The recent literature has intensively studied two classes of nonlocal
variational problems, namely the ones related to the minimisation of energy
functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the
ones related to the minimisation of fractional perimeters that act on
measurable sets of the Euclidean space.
  In this article, we relate these two types of variational problems. In
particular, we investigate the connection between the nonlocal minimal surfaces
and the minimisers of the $W^{s,1}$-seminorm.
  In particular, we show that a function is a minimiser for the fractional
seminorm if and only if its level sets are minimisers for the fractional
perimeter, and that the characteristic function of a nonlocal minimal surface
is a minimiser for the fractional seminorm; we also provide an existence result
for minimisers of the fractional seminorm, an explicit non-uniqueness example
for nonlocal minimal surfaces, and a Yin-Yang result describing the full and
void patterns of nonlocal minimal surfaces.
