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Rectifiable Sets, Densities and Tangent Measures
Camillo De Lellis, University of Zürich, Zurich, Switzerland
A publication of the European Mathematical Society.
Zurich Lectures in Advanced Mathematics
2008; 134 pp; softcover
Volume: 7
ISBN-10: 3-03719-044-2
ISBN-13: 978-3-03719-044-9
List Price: US$34
Member Price: US$27.20
Order Code: EMSZLEC/7
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The characterization of rectifiable sets through the existence of densities is a pearl of geometric measure theory. The difficult proof, due to Preiss, relies on many beautiful and deep ideas and novel techniques. Some of them have already proven useful in other contexts, whereas others have not yet been exploited. These notes give a simple and short presentation of the former and provide some perspective of the latter.

This text emerged from a course on rectifiability given at the University of Zürich. It is addressed both to researchers and students; the only prerequisite is a solid knowledge in standard measure theory. The first four chapters give an introduction to rectifiable sets and measures in Euclidean spaces, covering classical topics such as the area formula, the theorem of Marstrand and the most elementary rectifiability criterions. The fifth chapter is dedicated to a subtle rectifiability criterion due to Marstrand and generalized by Mattila, and the last three focus on Preiss' result. The aim is to provide a self-contained reference for anyone interested in an overview of this fascinating topic.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Graduate students and research mathematicians interested in analysis.

Table of Contents

  • Introduction
  • Notation and preliminaries
  • Marstsand's Theorem and tangent measures
  • Rectifiability
  • The Marstrand-Mattila rectifiability criterion
  • An overview of Preiss' proof
  • Moments and uniqueness of the tangent measure at infinity
  • Flat versus curved at infinity
  • Flatness at infinity implies flatness
  • Open problems
  • Appendix A. Proof of Theorem 3.11
  • Appendix B. Gaussian integrals
  • Bibliography
  • Index
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