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

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