Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions
Abstract
Makeev [2] stated that every finite Borel measure in ℝd assigning zero mass to hyperplanes can be cut by d mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk--Ulam theorem for O(k): every Bk-equivariant map from O(k) to a natural representation of the hyperoctahedral group Bk has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].
Related
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- Finding solutions with the Borsuk-Ulam theorem
- Nullhomotopic equivariant shifts and Borsuk-Ulam-compatible compact abelian groups
- A question of Ulam and a theorem of de Rham
- Counterexamples and symmetry for uneven orthogonal mass partitions in the plane
BibTeX
@misc{musin2026borsukulam,
title = {{Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions}},
author = {Oleg R. Musin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09688},
eprint = {2609.09688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09688},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions
AU - Oleg R. Musin
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.09688
UR - https://arxiv.org/abs/2609.09688
AB - Makeev [2] stated that every finite Borel measure in ℝ^d assigning zero mass to hyperplanes can be cut by d mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk--Ulam theorem for O(k): every B_k-equivariant map from O(k) to a natural representation of the hyperoctahedral group B_k has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].
ER -