A new differentiation, shape of the unit ball and perimeter measure

Abstract

We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent groups. Our techniques are new also in the classical Euclidean framework. When the distance is sub-Riemannian, we are lead to an unexpected relationship between the validity of a suitable area formula with respect to a distance and the profile of its corresponding unit ball.