Characteristic points, rectifiability and perimeter measure on stratified groups

Magnani, Valentino. "Characteristic points, rectifiability and perimeter measure on stratified groups." Journal of the European Mathematical Society 008.4 (2006): 585-609. <http://eudml.org/doc/277231>.

@article{Magnani2006,
abstract = {We establish an explicit connection between the perimeter measure of an open set $E$ with $C^1$ boundary and the spherical Hausdorff measure $S^\{Q−1\}$ restricted to $\partial E$, 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 $S^\{Q−1\}(\partial E)$ 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 $S^\{Q−k\}$-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, called $(\mathbb \{G\},\mathbb \{R\}^k)$-rectifiability, and we prove that Euclidean $k$-codimensional rectifiable sets are $(\mathbb \{G\},\mathbb \{R\}^k)$-rectifiable.},
author = {Magnani, Valentino},
journal = {Journal of the European Mathematical Society},
keywords = {stratified groups; characteristic points; perimeter measure; Hausdorff measure; Hausdorff measure},
language = {eng},
number = {4},
pages = {585-609},
publisher = {European Mathematical Society Publishing House},
title = {Characteristic points, rectifiability and perimeter measure on stratified groups},
url = {http://eudml.org/doc/277231},
volume = {008},
year = {2006},
}

TY - JOUR
AU - Magnani, Valentino
TI - Characteristic points, rectifiability and perimeter measure on stratified groups
JO - Journal of the European Mathematical Society
PY - 2006
PB - European Mathematical Society Publishing House
VL - 008
IS - 4
SP - 585
EP - 609
AB - We establish an explicit connection between the perimeter measure of an open set $E$ with $C^1$ boundary and the spherical Hausdorff measure $S^{Q−1}$ restricted to $\partial E$, 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 $S^{Q−1}(\partial E)$ 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 $S^{Q−k}$-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, called $(\mathbb {G},\mathbb {R}^k)$-rectifiability, and we prove that Euclidean $k$-codimensional rectifiable sets are $(\mathbb {G},\mathbb {R}^k)$-rectifiable.
LA - eng
KW - stratified groups; characteristic points; perimeter measure; Hausdorff measure; Hausdorff measure
UR - http://eudml.org/doc/277231
ER -