  We consider the family of non-local and non-convex functionals proposed and
investigated by J. Bourgain, H. Brezis and H.-M. Nguyen in a series of papers
of the last decade. It was known that this family of functionals
Gamma-converges to a suitable multiple of the Sobolev norm or the total
variation, depending on the summability exponent, but the exact constants and
the structure of recovery families were still unknown, even in dimension one.
  We prove a Gamma-convergence result with explicit values of the constants in
any space dimension. We also show the existence of recovery families consisting
of smooth functions with compact support.
  The key point is reducing the problem first to dimension one, and then to a
finite combinatorial rearrangement inequality.
