We establish an explicit formula between the perimeter measure of an open set with boundary and the spherical Hausdorff measure restricted to , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of is less than or equal to up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo and Nhieu. The crucial ingredient of this result is the negligibility of ``characte\-ristic points" of the boundary. We introduce the notion of ``horizontal point", which extends the notion of characteristic point to arbitrary submanifolds and we prove that the set of horizontal points of a -codimensional submanifold is -negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, namely, -rectifiability and we prove that Euclidean -codimensional rectifiable sets are -rectifiable.
Cite this article
Valentino Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups. J. Eur. Math. Soc. 8 (2006), no. 4, pp. 585–609DOI 10.4171/JEMS/68