Year: 2026
Type: preprint
Abstract: In this article we study the analog of Kakeya needle problem for $(n-1)$-rectifiable sets in $\mathbb{R}^n$ and construct the related Nikodym type sets. The novelty of our approach lies in combining three ingredients: the two-dimensional Venetian blind-type construction for isometries in $\mathbb{R}^n$; the normal geometry of a $(n-1)$-rectifiable set viewed in the projective setting; and measure estimates of moving $(n-1)$-rectifiable sets by isometries in $\mathbb{R}^n$. Together, these ingredients enable us to move $(n-1)$-rectifiable sets along paths of isometries in $\mathbb{R}^n$ and cover a Lebesgue null set.
Source: arXiv (Cornell University)
Authors: Li, Wenjuan, Wang, Hongqiang, Wang, Huiju
Institutions: Northwestern Polytechnical University, Henan University
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G2371208817
OpenAlex ID: W7202182731
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