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An orthogonality inequality in ℝℙn−1 and the double-cap conjecture

  • Allen Ibiapina
  • Carlos Gomes
  • Alan Pio
  • Samuel Belo
  • Ernandes Ferreira
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Abstract

The orthogonal neighborhood of a set in real projective space is the union of the hyperplanes perpendicular to its points. We prove that caps minimize neighborhood measure among measurable sets of fixed measure in ℝℙn−1, for n ≥ 3, with rigidity up to null sets. This yields the double-cap conjecture: a measurable subset of Sn−1 containing no orthogonal pair has measure at most that of two opposite open caps of radius π/4. The argument combines compactness of the Funk transform, rotation comparisons, and projective variations. We also obtain qualitative stability and determine the measurable fractional chromatic number of the spherical orthogonality graph. AI played an assistive role in preparing this paper. The formalization and verification workflow for a separate Lean 4 proof of the main inequality and double-cap result, based on a different argument, were carried out almost entirely using AI-assisted tools.

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Cite

BibTeXRIS
BibTeX
@misc{ibiapina2026orthogonality,
  title = {{An orthogonality inequality in $\mathbbRP^n-1$ and the double-cap conjecture}},
  author = {Allen Ibiapina and Carlos Gomes and Alan Pio and Samuel Belo and Ernandes Ferreira},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27523},
  eprint = {2610.27523},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27523},
}
RIS
TY  - UNPB
TI  - An orthogonality inequality in ℝℙ^n−1 and the double-cap conjecture
AU  - Allen Ibiapina
AU  - Carlos Gomes
AU  - Alan Pio
AU  - Samuel Belo
AU  - Ernandes Ferreira
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27523
UR  - https://arxiv.org/abs/2610.27523
AB  - The orthogonal neighborhood of a set in real projective space is the union of the hyperplanes perpendicular to its points. We prove that caps minimize neighborhood measure among measurable sets of fixed measure in ℝℙ^n−1, for n ≥ 3, with rigidity up to null sets. This yields the double-cap conjecture: a measurable subset of S^n−1 containing no orthogonal pair has measure at most that of two opposite open caps of radius π/4. The argument combines compactness of the Funk transform, rotation comparisons, and projective variations. We also obtain qualitative stability and determine the measurable fractional chromatic number of the spherical orthogonality graph. AI played an assistive role in preparing this paper. The formalization and verification workflow for a separate Lean 4 proof of the main inequality and double-cap result, based on a different argument, were carried out almost entirely using AI-assisted tools.
ER  -