Optimal constants for a non-local approximation of Sobolev norms and total variation

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.