The rectifiable subsets of Euclidean space
Abstract: In this paper the structure of a subset with 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 dimensional smooth submanifold of with nonzero Gaussian curvature at every point. If with , then there exists a countably 1-rectifiable Borel subset R of such that is purely unrectifiable and for almost all .
Furthermore, if in addition E is measurable and for almost all then . Consequently, E is purely unrectifiable.
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Keywords: Geometric measure theory, structure theory, Gaussian curvature, countably k-rectifiable, purely unrectifiable, k-dimensional Hausdorff measure, Suslin sets, k-dimensional upper density of at u
Article copyright: © Copyright 1978 American Mathematical Society