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A counterexample to Gale's conjecture in ℝ3

  • Vsevolod Voronov
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Abstract

It is known that for every partition of the sphere S2 of unit diameter into four parts, the largest diameter of a part is at least √((3+√3)/6) = 0.88807…, and this bound is attained. The conjecture of D.~Gale (1953) asserts that no set in ℝ3 has a larger ratio. In this paper, we construct a $101$-point set V⊂ℝ3 such that every partition of V into four parts contains a part of diameter greater than 0.88928 diam V.

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Cite

BibTeXRIS
BibTeX
@misc{voronov2026counterexample,
  title = {{A counterexample to Gale's conjecture in $\mathbbR^3$}},
  author = {Vsevolod Voronov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28828},
  eprint = {2610.28828},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28828},
}
RIS
TY  - UNPB
TI  - A counterexample to Gale's conjecture in ℝ^3
AU  - Vsevolod Voronov
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28828
UR  - https://arxiv.org/abs/2610.28828
AB  - It is known that for every partition of the sphere S^2 of unit diameter into four parts, the largest diameter of a part is at least √((3+√3)/6) = 0.88807…, and this bound is attained. The conjecture of D.~Gale (1953) asserts that no set in ℝ^3 has a larger ratio. In this paper, we construct a $101$-point set V⊂ℝ^3 such that every partition of V into four parts contains a part of diameter greater than 0.88928 diam V.
ER  -