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We consider the approximation of the total variation of a function by thefamily of non-local and non-convex functionals introduced by H. Brezis andH.-M. Nguyen in a recent paper. The approximating functionals are definedthrough double integrals in which every pair of points contributes according tosome interaction law.  In this paper we answer two open questions concerning the dependence of theGamma-limit on the interaction law. In the first result, we show that theGamma-limit depends on the full shape of the interaction law, and not only onthe values in a neighborhood of the origin. In the second result, we show thatthere do exist interaction laws for which the Gamma-limit coincides with thepointwise limit on smooth functions.  The key argument is that for some special classes of interaction laws thecomputation of the Gamma-limit can be reduced to studying the asymptoticbehavior of suitable multi-variable minimum problems.

On the gap between Gamma-limit and pointwise limit for a non-local approximation of the total variation