A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure The study of the geometry of n-uniform measures in $$\mathbb {R}^{d}$$ R d has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension $$n-3$$ n - 3 . In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most $$n-3$$ n - 3 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1206-9
Publisher site
See Article on Publisher Site

Abstract

The study of the geometry of n-uniform measures in $$\mathbb {R}^{d}$$ R d has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension $$n-3$$ n - 3 . In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most $$n-3$$ n - 3 .

Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 11, 2017

References

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