Recognition of inscribed polytopes in fixed dimension ≥ 4

Determine the computational complexity of recognizing inscribed polytopes (not necessarily simplicial) in fixed dimensions d ≥ 4.

Background

Inscribed polytopes are those with all vertices on a sphere; they correspond via stereographic projection to Delaunay subdivisions. While 3D recognition has polynomial algorithms (Rivin), higher fixed dimensions lack complexity classification.

The authors explicitly flag fixed dimensions ≥ 4 as open.

References

The complexity of the problem for inscribed polytopes (not necessarily simplicial) of constant dimension $d\geq 4$ is open as well.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Compendium — Problem 'Inscribed Simplicial Polytope Realizability'