Adler–Rivara–Levin non-degeneracy conjecture for Longest Edge Bisection

For every nondegenerate tetrahedron $T_0\subset\mathbb{R}^3$ and every resolution of ties between equally long edges, repeatedly bisecting a longest edge produces a family $\mathcal{R}(T_0)$ of tetrahedra that is uniformly nondegenerate: there exists a constant $c(T_0)>0$ such that $r(T)/\operatorname{diam}(T)\ge c(T_0)$ for every $T\in\mathcal{R}(T_0)$ , where $r(T)$ is the inradius and $\operatorname{diam}(T)$ is the diameter. A stronger formulation described in the source asserts an appropriate periodicity of the resulting orbit in projective shape space.

References

Primary source

arXiv

Additional references

Degenerating orbits of the Longest Edge Bisection process — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A 2026 preprint claims a counterexample showing that longest-edge bisection can produce increasingly flat tetrahedra, overturning the conjecture.

The Adler–Rivara–Levin conjecture asserts non-degeneracy for tetrahedra generated by longest-edge bisection, regardless of how ties between longest edges are resolved. A recent preprint claims this assertion is false and also reports further asymptotic and nonperiodic behavior.

Known results

Earlier numerical work suggested that longest-edge bisection does not produce degenerating tetrahedra and outlined a possible proof strategy.
The same work reported finite similarity orbits with lengths exceeding $37$ , including $38$ , $39$ , $42$ , $43$ , and $44$ , concerning a separate Adler orbit-length conjecture.

August 24, 2026 claimed counterexample

Sergey Korotov’s preprint The longest-edge bisection algorithm may produce degenerating tetrahedra, submitted on August 24, 2026, claims an explicit degenerating sequence. It says the sequence violates shape-regularity and both minimum- and maximum-angle conditions, implying that arbitrary longest-edge tie-breaking does not ensure non-degeneracy. The result is unverified.

Current status (as of September 2026): The conjecture has a claimed explicit counterexample, but its correctness and the resulting refutation remain unverified.

Sources

No solutions have been posted yet.