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Petty Equilateral-Set Conjecture

The Petty equilateral-set conjecture states that every real normed space of dimension d contains an equilateral set of at least d+1 points. The points need not be affinely independent, and the assertion does not require extension of a prescribed equilateral set (Petty 1971).

Makeev (2007) established the four-dimensional case. Yang (2026) reported the five-dimensional case, namely six equidistant points in every real five-dimensional normed space. Yang (2026) states that Tencent Hunyuan hyprover generated the proof and that he checked and rewrote it. The bordism and transversality argument requires independent specialist assessment. The general conjecture remains open.

See also

Equilateral Set, Normed Space

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References

Makeev, V. V. "Equilateral Simplices in Normed 4-Space." J. Math. Sci. (N.Y.) 140, 548-550, 2007. https://doi.org/10.1007/s10958-007-0436-3.Petty, C. M. "Equilateral Sets in Minkowski Spaces." Proc. Amer. Math. Soc. 29, 369-374, 1971. https://doi.org/10.1090/S0002-9939-1971-0275294-8.Yang, K. "Petty's Equilateral-Set Conjecture in Dimension Five." 29 Oct 2026. https://arxiv.org/abs/2610.39967.

Cite this as:

Weisstein, Eric W. "Petty Equilateral-Set Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PettyEquilateral-SetConjecture.html

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