Kusner's equilateral-set conjecture in finite-dimensional lp spaces

Kusner posed the question in 1983: whether every equilateral set in $\ell_p^n$, for $1<p<\infty$, has exactly $n+1$ points. The universal assertion is now known to fail in part of the range, but its precise behavior remains unsettled.

Known results

• The equality holds for $p=2$ in every dimension (classical).
• Swanepoel proved $e(\ell_4^n)=n+1$ for every $n$.
• Swanepoel constructed sets larger than $n+1$ for every fixed $1<p<2$ in sufficiently large dimension.
• For even integer $p$, earlier work gives explicit linear upper bounds, including $(2\lceil p/4\rceil-1)n+1$.

2026 partial results and counterexample claim

Ge, Xu, and Zhou claim $e(\ell_p^n)=n+1$ throughout $2\leq p\leq4$, with linear bounds on intervals $[4k+2,4k+4]$ and $O_p(n\log n)$ bounds on complementary intervals. A later preprint claims an exact $58$-point equilateral set in $\ell_5^{56}$, proving $e(\ell_5^{56})\geq58>57$ and hence disproving Kusner’s conjecture; the claim is not independently verified in the supplied sources.

Current status (as of August 2026): The conjecture is proved for $2\leq p\leq4$ and disproved for $1<p<2$ in sufficiently large dimensions; the claimed $\ell_5^{56}$ counterexample would extend disproof to $p>2$, but remains unverified, and the remaining exponent ranges are open.