On the union of essentially distinct $\delta$-tubes

We say two $\delta$-tubes (dimension $\delta\times\cdots\times\delta\times1$) in $\mathbb{R}^n$ are essentially distinct if the measure of their intersection is smaller than a half of a single $\delta$-tube. For a collection of essentially distinct $\delta$-tubes, we give the asymptotically sharp lo...

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Bibliographic Details
Main Author Ren, Qiuyu
Format Journal Article
LanguageEnglish
Published 16.08.2019
Subjects
Mathematics - Classical Analysis and ODEs
Mathematics - Metric Geometry
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Summary:We say two $\delta$-tubes (dimension $\delta\times\cdots\times\delta\times1$) in $\mathbb{R}^n$ are essentially distinct if the measure of their intersection is smaller than a half of a single $\delta$-tube. For a collection of essentially distinct $\delta$-tubes, we give the asymptotically sharp lower bound for the measure of their union. Then we characterize all sharp examples. We will give a new measurement of convexity based on the X-ray transform.
DOI:10.48550/arxiv.1908.06000