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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The $ (\varphi , 1)$ rectifiable subsets of Euclidean space


Author: Samir Kar
Journal: Trans. Amer. Math. Soc. 237 (1978), 353-371
MSC: Primary 49F20
MathSciNet review: 0487725
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Abstract: In this paper the structure of a subset $ E \subset {{\mathbf{R}}^n}$ with $ {{\mathbf{H}}^1}(E) < \infty $ has been studied by examining its intersection with various translated positions of a smooth hypersurface B. The following result has been established:

Let B be a proper $ (n - 1)$ dimensional smooth submanifold of $ {{\mathbf{R}}^n}$ with nonzero Gaussian curvature at every point. If $ E \subset {{\mathbf{R}}^n}$ with $ {{\mathbf{H}}^1}(E) < \infty $, then there exists a countably 1-rectifiable Borel subset R of $ {{\mathbf{R}}^n}$ such that $ (E \sim R)$ is purely $ ({{\mathbf{H}}^1},1)$ unrectifiable and $ (E \sim R) \cap (g + B) = \emptyset $ for almost all $ g \in {{\mathbf{R}}^n}$.

Furthermore, if in addition E is $ {{\mathbf{H}}^1}$ measurable and $ E \cap (g + B) = \emptyset $ for $ {{\mathbf{H}}^n}$ almost all $ g \in {{\mathbf{R}}^n}$ then $ {{\mathbf{H}}^1}(E \cap R) = 0$. Consequently, E is purely $ ({{\mathbf{H}}^1},1)$ unrectifiable.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0487725-X
PII: S 0002-9947(1978)0487725-X
Keywords: Geometric measure theory, structure theory, Gaussian curvature, countably k-rectifiable, purely $ ({{\mathbf{H}}^k},k)$ unrectifiable, k-dimensional Hausdorff measure, Suslin sets, k-dimensional upper density of $ \varphi $ at u
Article copyright: © Copyright 1978 American Mathematical Society