For every integer n≥ 1 and every real p satisfying 2<p<∞, the maximum cardinality of an equilateral set in ell_p^n is e(ell_p^n)=n+1. Equivalently, every set X⊂R^n whose distinct points have pairwise ell_p-distance 1 has at most n+1 elements.

Kusner’s conjecture on equilateral sets in ℓ_p^n

R. B. Kusner posed the conjecture in 1983: every equilateral set in $\ell_p^n$ should have at most $n+1$ points for $p\ge 2$ . The new construction directly challenges this finite- $p$ range.

Known results

The conjecture was known for $p=2$ and $p=4$ before the 2026 developments.
For every $1<p<2$ , earlier constructions give failures in sufficiently large dimension; some explicit cases already occur in dimension $6$ (2003–2004).
Large- $p$ work provides improved upper bounds but does not settle the conjecture (2020).

June–August 2026 developments

A June preprint claims the bound $e(\ell_p^n)\le n+1$ for all $2\le p\le4$ . In August, Logan R. Chalmers claimed an exact-arithmetic-certified configuration with $e(\ell_5^{56})\ge58>57$ , the first finite- $p\ge2$ example exceeding $n+1$ ; persistence near $p=5$ would place the failure threshold in $[4,5)$ .

Current status (as of August 2026): The bound is claimed for $2\le p\le4$ and a counterexample is claimed at $p=5$ , but both 2026 preprint claims remain unverified.

Kusner’s conjecture on equilateral sets in ℓ_p^n

Progress summary

Primary source

Additional references

Solutions 0