Dice Question Streamline Icon: https://streamlinehq.com

Simplicial Polytopes in Fixed Dimension ≥ 4

Classify the computational complexity of recognizing realizability of simplicial polytopes in fixed constant dimension d ≥ 4.

Information Square Streamline Icon: https://streamlinehq.com

Background

Realizability of polytopes is a canonical ER-complete problem, and strong universality holds in various settings.

While 4-dimensional realizability is known ER-complete, the complexity for fixed higher-dimensional simplicial polytopes is explicitly stated as an open problem.

References

The complexity of the problem for simplicial polytopes of constant dimension $d\geq 4$ is open, see Richter-Gebert.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “4-dimensional Polytope Realizability” entry