A Study on Kakeya Needle Problem for $(n-1)$-Rectifiable Sets

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.

45 pages,3 figures