Delaunay Triangulation Recognition in Fixed Dimension ≥ 3

Determine the computational complexity of recognizing Delaunay triangulations in fixed constant dimension d ≥ 3.

Delaunay triangulations correspond to inscribed simplicial polytopes via stereographic projection and are ER-complete in general settings.

The survey explicitly identifies the constant-dimension case (d ≥ 3) as an open problem.

The complexity of the problem in constant dimension $d\geq 3$ is open.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “Delaunay Triangulation” entry