Sharper factor-one-half diameter decay in higher-dimensional longest-edge bisection

Determine whether repeated longest-edge bisection in dimensions greater than two satisfies a diameter-decay estimate of the form $h_{kd}\le C2^{-k}$ for a constant $C$ depending on the initial simplex.

Background

Stynes proved a diameter-decay estimate with factor (3/2)k(\sqrt{3}/2)^k over blocks of dd bisection levels and, based on computations, raised the possibility of the sharper planar-type factor $1/2$ in dimensions greater than two.

The paper explains that this sharper estimate would imply non-degeneracy, but its tetrahedral counterexample for bisection rules it out for arbitrary tetrahedra under the permitted edge choices. The problem is included because the paper explicitly says that Stynes posed it.

References

He proved a bound of the form $h_{kd}\le C(\sqrt3/2)k$ and, on the basis of computations, posed the problem of obtaining the sharper factor $1/2$ for dimensions $d>2$, namely $h_{kd}\le C2{-k}$, as already known for triangles.

— On degeneration of tetrahedra under longest-edge n-section refinement  (2609.29153 - Korotov, 24 Sep 2026) in Section 1, subsection “Historical background”