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Väisälä quasihyperbolic geodesic and convexity conjectures

Let D⊊RnD\subsetneq\mathbb{R}^n be a convex domain, let x∈Dx\in D, and let r≤arctan⁡(2)/2r\leq \arctan(\sqrt{2})/\sqrt{2}. Define the quasihyperbolic metric by kD(x,y)=inf⁡γ∫γ∣dz∣dist⁡(z,∂D)k_D(x,y)=\inf_\gamma\int_\gamma \frac{\lvert dz\rvert}{\operatorname{dist}(z,\partial D)}, where the infimum is over rectifiable curves γ⊂D\gamma\subset D joining xx to yy, and define the quasihyperbolic ball by BkD(x,r)={y∈D:kD(x,y)<r}B_{k_D}(x,r)=\{y\in D:k_D(x,y)<r\}. The conjecture asserts that BkD(x,r)B_{k_D}(x,r) is Euclidean convex.

References

Primary source

arXiv

Additional references

geometric-function-theorymetric-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to settle the quasihyperbolic geodesic and convexity conjectures, but the claim has not been independently verified.

The conjectures concern existence, uniqueness, prolongation, smoothness, and convexity properties of quasihyperbolic geodesics and balls in Banach-space domains. Earlier work left Väisälä-related questions open, while also producing negative results in some nonreflexive settings.

Known results

  • Gehring and Osgood proved existence in domains of Rn\mathbb{R}^n; Martin proved smoothness.
  • Martio and Väisälä established existence and uniqueness in convex domains of uniformly convex Banach spaces.
  • Rasila and Talponen extended uniqueness to convex domains of reflexive strictly convex Banach spaces and gave a nonreflexive counterexample to existence.
  • Their work records Väisälä’s open characterization of reflexivity via geodesicity of open half-spaces.

September 2026 CAT(2)\mathrm{CAT}(2) preprint

On September 21, 2026, Toni Ikonen and Abhishek Pandey’s preprint Quasihyperbolic domains are CAT(2) was reported as proving the curvature theorem and deriving the stated geodesic uniqueness, prolongation, convexity, and sphere-regularity conclusions. This is a claimed complete resolution, not yet independently verified.

Current status (as of September 2026): Ikonen and Pandey claim that the listed conjectures are resolved, but independent verification is pending, so the problem remains unconfirmed rather than settled.

Sources

Solutions 0

No solutions have been posted yet.