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We consider the family of non-local and non-convex functionals introduced byH. Brezis and H.-M. Nguyen in a recent paper. These functionals Gamma-convergeto a multiple of the Sobolev norm or the total variation, depending on asummability exponent, but the exact values of the constants are unknown in manycases.  We describe a new approach to the Gamma-convergence result that leads in somespecial cases to the exact value of the constants, and to the existence ofsmooth recovery families.

On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation