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A diameter-one set in ℝ5 that cannot be covered by six diameter-one balls

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

We construct a compact set of diameter $1$ in ℝ5 which cannot be covered by six Euclidean balls of diameter $1$. Its convex hull is a full-dimensional convex body satisfying the same property. Moreover, a finite subset of diameter $1$ cannot be covered by six closed balls even when their radius is increased to $0.50001$. The compact set is the quadratic image of the unit sphere in the space of symmetric 3×3 matrices of trace zero. A hypothetical cover leads to twelve spherical ellipses covering 𝕊2 and arranged in six antipodal pairs. Using Euler's formula, we replace them by twelve pentagonal regions contained in the ellipses. An area estimate then gives a contradiction.

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Cite

BibTeXRIS
BibTeX
@misc{arman2026diameterone,
  title = {{A diameter-one set in $\mathbbR^5$ that cannot be covered by six diameter-one balls}},
  author = {A. Arman and A. Bondarenko and A. Prymak and D. Radchenko},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08198},
  eprint = {2611.08198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08198},
}
RIS
TY  - UNPB
TI  - A diameter-one set in ℝ^5 that cannot be covered by six diameter-one balls
AU  - A. Arman
AU  - A. Bondarenko
AU  - A. Prymak
AU  - D. Radchenko
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08198
UR  - https://arxiv.org/abs/2611.08198
AB  - We construct a compact set of diameter $1$ in ℝ^5 which cannot be covered by six Euclidean balls of diameter $1$. Its convex hull is a full-dimensional convex body satisfying the same property. Moreover, a finite subset of diameter $1$ cannot be covered by six closed balls even when their radius is increased to $0.50001$. The compact set is the quadratic image of the unit sphere in the space of symmetric 3×3 matrices of trace zero. A hypothetical cover leads to twelve spherical ellipses covering 𝕊^2 and arranged in six antipodal pairs. Using Euler's formula, we replace them by twelve pentagonal regions contained in the ellipses. An area estimate then gives a contradiction.
ER  -