A counterexample to Kusner's conjecture on equilateral sets

@misc{chalmers2026counterexample,
  title = {{A counterexample to Kusner's conjecture on equilateral sets}},
  author = {Logan R. Chalmers},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.14013},
  eprint = {2608.14013},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14013},
}

TY  - UNPB
TI  - A counterexample to Kusner's conjecture on equilateral sets
AU  - Logan R. Chalmers
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.14013
UR  - https://arxiv.org/abs/2608.14013
AB  - We disprove Kusner's 1983 conjecture that every equilateral set in ℓ_p^n with 2<p<∞ has at most n+1 points: there exist $58$ points in ℝ^56 whose pairwise ℓ_5 distances are all equal, so the maximum equilateral-set size satisfies e(ℓ_5^56)≥58>57. This is the first equilateral set of more than n+1 points in ℓ_p^n for any finite p≥2. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for 2≤ p≤4, the infimum of exponents at which it fails lies in [4,5). The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.
ER  -