Kusner’s equilateral-set conjecture in l_p spaces

Kusner’s equilateral-set question predicts a dimension-plus-one maximum in ℓₚ. For every p>4, the counterexample target is an equilateral set of 8m points in ℓₚ^(8m−2) for an appropriate power-of-two m.

Why it matters and what counts as progress

Why it matters: It tests the proposed cardinality bound across the remaining p>4 range.

What counts as progress: An equilateral configuration exceeding dimension plus one in the remaining p range, or a sharp bound for a class where the conjectured maximum survives.

How the problem developed

1983 · Conjecture [1]
Robert Kusner Conjectured that for 1<p<∞, the largest equilateral set in ℓ_pⁿ has n+1 points; the ℓ₁ case predicts 2n.
2 Jun 2026 · Proof in range [1]
Hong-Jun Ge, Zixiang Xu and Yang Zhou Proved the conjectured n+1 bound in every dimension for 2≤p≤4.
13 Sept 2026 · Counterexamples [2]
Nathan Xiong Disproved the n+1 conjecture for every p>4 by constructing 8m equilateral points in ℓ_p^(8m−2).

AI contribution

Kusner's conjecture is false for p > 4

Nathan Xiong

For every real p>4, Xiong constructs an equilateral set of 8m points in ℓₚ^(8m−2) for some power-of-two m, exceeding the proposed dimension-plus-one maximum.

AI involvementAI-centered

ImportanceMajor

Technical verificationReported

Human understandingFull