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Peer-reviewedCC BY-NCBulletin of the London Mathematical Society

On asymptotic Lebesgue's universal covering problem

  • A. Arman
  • A. Bondarenko
  • A. Prymak
  • Danylo Radchenko
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Abstract

A universal cover in 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 (). A simple universal cover in is provided by the classical theorem of Jung, which states that any set of diameter 1 in an ‐dimensional Euclidean space is contained in a ball of radius ; in other words, is a universal cover in . We show that in high dimensions, Jung's ball is asymptotically optimal with respect to the volume, namely, for any universal cover ,

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Cite

BibTeXRIS
BibTeX
@article{arman2026asymptotic,
  title = {{On asymptotic Lebesgue's universal covering problem}},
  author = {A. Arman and A. Bondarenko and A. Prymak and Danylo Radchenko},
  journal = {Bulletin of the London Mathematical Society},
  year = {2026},
  month = oct,
  doi = {10.1112/blms.70514},
  eprint = {2512.04023},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05851837v1},
  note = {CC BY-NC},
}
RIS
TY  - JOUR
TI  - On asymptotic Lebesgue's universal covering problem
AU  - A. Arman
AU  - A. Bondarenko
AU  - A. Prymak
AU  - Danylo Radchenko
PY  - 2026
DA  - 2026/10/07
JO  - Bulletin of the London Mathematical Society
DO  - 10.1112/blms.70514
UR  - https://hal.science/hal-05851837v1
AB  - A universal cover in 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 (). A simple universal cover in is provided by the classical theorem of Jung, which states that any set of diameter 1 in an ‐dimensional Euclidean space is contained in a ball of radius ; in other words, is a universal cover in . We show that in high dimensions, Jung's ball is asymptotically optimal with respect to the volume, namely, for any universal cover ,
SN  - 0024-6093
SN  - 1469-2120
ER  -