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Polygons and coplanar points on submanifolds of codimension 2
Abstract
Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in ℝn must meet some affine two-plane in five points when n is large. In this paper, we answer this question for every n ≥ 6. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound n ≥ 6 is sharp for affinely regular pentagons. More generally, for every prime p ≥ 5, the graph of a continuous map from a ball in ℝp−1 to ℝ2 contains an affinely regular p-gon. A second argument gives six coplanar points when n ≥ 7. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.
Related
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BibTeX
@misc{sohail2026polygons,
title = {{Polygons and coplanar points on submanifolds of codimension 2}},
author = {Sohail and Sarkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14032},
eprint = {2610.14032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14032},
}
RIS
TY - UNPB
TI - Polygons and coplanar points on submanifolds of codimension 2
AU - Sohail
AU - Sarkar
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.14032
UR - https://arxiv.org/abs/2610.14032
AB - Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in ℝ^n must meet some affine two-plane in five points when n is large. In this paper, we answer this question for every n ≥ 6. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound n ≥ 6 is sharp for affinely regular pentagons. More generally, for every prime p ≥ 5, the graph of a continuous map from a ball in ℝ^p−1 to ℝ^2 contains an affinely regular p-gon. A second argument gives six coplanar points when n ≥ 7. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.
ER -