On asymptotic Lebesgue's universal covering problem
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914178939224064 |
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| author | Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo |
| author_facet | Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo |
| contents | Universal cover in $\mathbb{E}^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ($n=2$).
A simple universal cover in $\mathbb{E}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{\tfrac{n}{2n+2}}$; in other words, $J_n$ is a universal cover in $\mathbb{E}^n$.
We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On asymptotic Lebesgue's universal covering problem Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo Metric Geometry Combinatorics Primary 52C17, Secondary 52A20, 52A22, 52A38, 52A39, 52A40, 49Q20 Universal cover in $\mathbb{E}^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ($n=2$). A simple universal cover in $\mathbb{E}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{\tfrac{n}{2n+2}}$; in other words, $J_n$ is a universal cover in $\mathbb{E}^n$. We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$ |
| title | On asymptotic Lebesgue's universal covering problem |
| topic | Metric Geometry Combinatorics Primary 52C17, Secondary 52A20, 52A22, 52A38, 52A39, 52A40, 49Q20 |
| url | https://arxiv.org/abs/2512.04023 |