Kakeya problem and projection problem for $k$-geodesics in Grassmannians
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911834841284608 |
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| author | Gan, Shengwen |
| author_facet | Gan, Shengwen |
| contents | The Kakeya problem in $\mathbb{R}^n$ is about estimating the size of union of $k$-planes; the projection problem in $\mathbb{R}^n$ is about estimating the size of projection of a set onto every $k$-plane ($1\le k\le n-1$). The $k=1$ case has been studied on general manifolds in which $1$-planes become geodesics, while $k\ge 2$ cases were still only considered in $\mathbb{R}^n$. We formulate these problems on homogeneous spaces, where $k$-planes are replaced by $k$-dimensional totally geodesic submanifolds. After formulating the problem, we prove a sharp estimate for Grassmannians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kakeya problem and projection problem for $k$-geodesics in Grassmannians Gan, Shengwen Classical Analysis and ODEs Combinatorics Metric Geometry The Kakeya problem in $\mathbb{R}^n$ is about estimating the size of union of $k$-planes; the projection problem in $\mathbb{R}^n$ is about estimating the size of projection of a set onto every $k$-plane ($1\le k\le n-1$). The $k=1$ case has been studied on general manifolds in which $1$-planes become geodesics, while $k\ge 2$ cases were still only considered in $\mathbb{R}^n$. We formulate these problems on homogeneous spaces, where $k$-planes are replaced by $k$-dimensional totally geodesic submanifolds. After formulating the problem, we prove a sharp estimate for Grassmannians. |
| title | Kakeya problem and projection problem for $k$-geodesics in Grassmannians |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2404.04290 |