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Orthogonal Pairs in Maps from the Sphere to the Circle
Abstract
We prove that, for any f:S2→ S1 and any ε>0, there exist orthogonal vectors x,y∈ S2 such that the length of the shortest arc between f(x) and f(y) is at most π/2 +ε. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.
Related
- Sobolev Orthogonal Polynomials on the unit circle with varying inner product
- The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity
- The number of touching pairs of congruent sphere packings in Euclidean 3-space
- Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
- Nielsen coincidence theory of (n,m)-valued pairs of maps
BibTeX
@misc{noel2026orthogonal,
title = {{Orthogonal Pairs in Maps from the Sphere to the Circle}},
author = {Jonathan A. Noel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22680},
eprint = {2609.22680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22680},
}
RIS
TY - UNPB
TI - Orthogonal Pairs in Maps from the Sphere to the Circle
AU - Jonathan A. Noel
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22680
UR - https://arxiv.org/abs/2609.22680
AB - We prove that, for any f:S^2→ S^1 and any ε>0, there exist orthogonal vectors x,y∈ S^2 such that the length of the shortest arc between f(x) and f(y) is at most π/2 +ε. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.
ER -