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PreprintCC BY 4.0arXiv

Orthogonal partition by four hyperplanes

  • Alexey Fakhrutdinov
  • Platon Khamatov
  • Oleg R. Musin
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Abstract

We study orthogonal (4,3)-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number m of measures, we determine the exact dimension threshold of Musin's polynomial criterion and express it in terms of the binary expansion of m. The threshold coincides with the geometric lower bound precisely when m=2j−1. Consequently, Δ*(2j−1,4,3)=7·2j−1−2 for every j≥1.

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Cite

BibTeXRIS
BibTeX
@misc{fakhrutdinov2026orthogonal,
  title = {{Orthogonal partition by four hyperplanes}},
  author = {Alexey Fakhrutdinov and Platon Khamatov and Oleg R. Musin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19214},
  eprint = {2610.19214},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19214},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Orthogonal partition by four hyperplanes
AU  - Alexey Fakhrutdinov
AU  - Platon Khamatov
AU  - Oleg R. Musin
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.19214
UR  - https://arxiv.org/abs/2610.19214
AB  - We study orthogonal (4,3)-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number m of measures, we determine the exact dimension threshold of Musin's polynomial criterion and express it in terms of the binary expansion of m. The threshold coincides with the geometric lower bound precisely when m=2^j−1. Consequently, Δ^*(2^j−1,4,3)=7·2^j−1−2 for every j≥1.
ER  -