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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
DOI: https://doi.org/10.1090/S0002-9947-1978-0487725-X
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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Keywords: Geometric measure theory, structure theory, Gaussian curvature, countably <I>k</I>-rectifiable, purely <!– MATH $({{\mathbf {H}}^k},k)$ –> <IMG WIDTH="69" HEIGHT="45" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$({{\mathbf {H}}^k},k)$"> unrectifiable, <I>k</I>-dimensional Hausdorff measure, Suslin sets, <I>k</I>-dimensional upper density of <IMG WIDTH="19" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$\varphi$"> at <I>u</I>
Article copyright: © Copyright 1978 American Mathematical Society