Geodesic
Characteristic points, rectifiability and perimeter measure on stratified groups
Journal of the European Mathematical Society, Tome 8 (2006) no. 4, pp. 585-609

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We establish an explicit formula between the perimeter measure of an open set E with C1 boundary and the spherical Hausdorff measure SQ−1 restricted to derE, when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to SQ−1(derE) 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 “characteristic 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 k-codimensional submanifold is SQ−k-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, namely, (G,Rk)-rectifiability and we prove that Euclidean k-codimensional rectifiable sets are (G,Rk)-rectifiable.
DOI : 10.4171/jems/68
Classification : 22-XX, 28-XX, 53-XX, 00-XX
Keywords: stratified groups, characteristic points, perimeter measure, Hausdorff measure 
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     author = {Valentino Magnani},
     title = {Characteristic points, rectifiability and perimeter measure on stratified groups},
     journal = {Journal of the European Mathematical Society},
     pages = {585--609},
     publisher = {mathdoc},
     volume = {8},
     number = {4},
     year = {2006},
     doi = {10.4171/jems/68},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/68/}
}
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Valentino Magnani. Characteristic points, rectifiability and perimeter measure on stratified groups. Journal of the European Mathematical Society, Tome 8 (2006) no. 4, pp. 585-609. doi: 10.4171/jems/68

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