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arXiv · 2608.14013

A counterexample to Kusner's conjecture on equilateral sets

Logan R. Chalmers
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Abstract

We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2 57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, 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.

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Logan R. Chalmers. 2026-09-05. A counterexample to Kusner's conjecture on equilateral sets. https://arxiv.org/abs/2608.14013

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