Hausdorff volume in non equiregular sub-Riemannian manifolds

R. Ghezzi; F. Jean

doi:10.1016/j.na.2015.06.011

Hausdorff volume in non equiregular sub-Riemannian manifolds

R. Ghezzi, F. Jean

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).

Original language	English
Article number	10562
Pages (from-to)	345-377
Number of pages	33
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	126
DOIs	https://doi.org/10.1016/j.na.2015.06.011
Publication status	Published - 26 Oct 2015
Externally published	Yes

Keywords

Geometric measure theory
Hausdorff measures
Intrinsic volumes
Sub-Riemannian geometry

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10.1016/j.na.2015.06.011

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title = "Hausdorff volume in non equiregular sub-Riemannian manifolds",

abstract = "In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).",

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AU - Ghezzi, R.

AU - Jean, F.

PY - 2015/10/26

Y1 - 2015/10/26

N2 - In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).

AB - In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).

KW - Geometric measure theory

KW - Hausdorff measures

KW - Intrinsic volumes

KW - Sub-Riemannian geometry

U2 - 10.1016/j.na.2015.06.011

DO - 10.1016/j.na.2015.06.011

M3 - Article

AN - SCOPUS:84940451745

SN - 0362-546X

VL - 126

SP - 345

EP - 377

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

M1 - 10562

ER -

Hausdorff volume in non equiregular sub-Riemannian manifolds

Abstract

Keywords

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