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Conjecture on the Gromov–Hausdorff distance between consecutive spheres

Let D54AmD54A^m denote the unit mm-sphere with its intrinsic spherical metric, let dGH⁡d_{\operatorname{GH}} be the Gromov–Hausdorff distance, and let D701mD701_m be the quantity defined earlier in the paper through the lower-bound construction. Conjecture on consecutive spheres. For all m∈Nm\in\mathbb{N},

dGH⁡(Sm,Sm+1)=12ζm.d_{\operatorname{GH}}(\mathbb{S}^m,\mathbb{S}^{m+1})=\frac{1}{2}\zeta_m.

Theorem and explicit correspondence constructions in the paper establish the matching formula in several low-dimensional cases, including consecutive spheres in dimensions treated by the stated propositions; the conjecture asks whether the same exact value holds for every mm and remains open in general.

Sources & referencesView supporting material

Primary source

Sunhyuk Lim, Facundo Mémoli and Zane Smith, “The Gromov-Hausdorff distance between spheres”, arXiv:2105.00611 (2022).

gromov-hausdorff-distancemetric-geometryspheres

Progress summary

Refreshed
Partially solved

The conjecture is proved for the first three consecutive sphere pairs, but remains open in higher dimensions.

Lim, Mémoli, and Smith conjectured that the exact distance between consecutive unit spheres is the proposed lower bound for every dimension. The original work proves the first two cases and leaves the all-dimensional statement open.

Known results

  • The general lower bound is dGH⁡(Sm,Sm+1)≥12ζmd_{\operatorname{GH}}(\mathbb{S}^{m},\mathbb{S}^{m+1})\geq\frac{1}{2}\zeta_m.
  • The cases m=1m=1 and m=2m=2 are proved exactly in the original work.
  • The original work gives only a weaker general upper bound, with parity-dependent quantity ηm\eta_m.

2024 proof of the m=3m=3 case

A 2024 paper proves dGH⁡(S3,S4)=12arccos⁡(−1/4)=12ζ3d_{\operatorname{GH}}(\mathbb{S}^{3},\mathbb{S}^{4})=\frac{1}{2}\arccos(-1/4)=\frac{1}{2}\zeta_3, using computer-assisted distortion estimates. It suggests possible extensions to m=4,5,6m=4,5,6 but does not claim the general conjecture.

Current status (as of August 2026): The equality is settled for m=1,2,3m=1,2,3, while the conjecture for m≥4m\geq 4 remains open, with no retrieved proof or counterexample.

Sources

Solutions 0

No solutions have been posted yet.