 for every open set formula_20 we write formula_28 for formula_29.In his papers and , Ennio de Giorgi introduces the following smoothing operator, analogous to the Weierstrass transform in the one-dimensional case

As one can easily prove, formula_31 is a smooth function for all formula_32, such that

also, its gradient is everywhere well defined, and so is its absolute value

Having defined this function, De Giorgi gives the following definition of perimeter