PreprintCC BY 4.0arXiv
Kusner's conjecture is false for p>4
Abstract
An equilateral set is a set of points in a metric space whose pairwise distances are all equal. Kusner conjectured that the maximum cardinality of an equilateral set in ℝn with the ℓp metric is n+1 for every 1<p<∞. We disprove the conjecture for all p>4. In particular, for every such p, we define m=m(p) and construct an equilateral set of 8m points in ℝ8m−2. Previously, Swanepoel disproved the conjecture for 1<p<2, while Ge, Xu, and Zhou showed that it holds for 2≤ p≤ 4. Combining these prior results with the paper's main theorem resolves Kusner's conjecture for all 1<p<∞.
Related
BibTeX
@misc{xiong2026kusners,
title = {{Kusner's conjecture is false for $p>4$}},
author = {Nathan Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14794},
eprint = {2609.14794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14794},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kusner's conjecture is false for p>4
AU - Nathan Xiong
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14794
UR - https://arxiv.org/abs/2609.14794
AB - An equilateral set is a set of points in a metric space whose pairwise distances are all equal. Kusner conjectured that the maximum cardinality of an equilateral set in ℝ^n with the ℓ_p metric is n+1 for every 1<p<∞. We disprove the conjecture for all p>4. In particular, for every such p, we define m=m(p) and construct an equilateral set of 8m points in ℝ^8m−2. Previously, Swanepoel disproved the conjecture for 1<p<2, while Ge, Xu, and Zhou showed that it holds for 2≤ p≤ 4. Combining these prior results with the paper's main theorem resolves Kusner's conjecture for all 1<p<∞.
ER -