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An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound

  • Shuai Zeng
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Abstract

Posed by Lebesgue in 1914, the universal covering problem asks for the smallest-area planar convex set containing a congruent copy of every set of diameter at most one. We introduce an exact Reuleaux-type variational hierarchy for this constant: its monotone finite-arc values ΛM satisfy aLeb=limM→∞ΛM, and each level is a continuous finite-dimensional problem. We prove 0≤ aLeb−ΛM≤ C M−2, giving a controlled finite-arc route to the constant itself. As a certified low-order realization, an outward-rounded interval certificate for a regular finite Reuleaux subtest proves aLeb≥0.834, improving the lower-bound benchmark established by Brass and Sharifi in 2005.

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Cite

BibTeXRIS
BibTeX
@misc{zeng2026exact,
  title = {{An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound}},
  author = {Shuai Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01284},
  eprint = {2609.01284},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01284},
}
RIS
TY  - UNPB
TI  - An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound
AU  - Shuai Zeng
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01284
UR  - https://arxiv.org/abs/2609.01284
AB  - Posed by Lebesgue in 1914, the universal covering problem asks for the smallest-area planar convex set containing a congruent copy of every set of diameter at most one. We introduce an exact Reuleaux-type variational hierarchy for this constant: its monotone finite-arc values Λ_M satisfy a_Leb=lim_M→∞Λ_M, and each level is a continuous finite-dimensional problem. We prove 0≤ a_Leb−Λ_M≤ C M^−2, giving a controlled finite-arc route to the constant itself. As a certified low-order realization, an outward-rounded interval certificate for a regular finite Reuleaux subtest proves a_Leb≥0.834, improving the lower-bound benchmark established by Brass and Sharifi in 2005.
ER  -