Steinitz problem

The Steinitz problem is a algorithmic decision problem about polytopes. The problem is to determine if a given finite abstract polytope has a realization as a convex polytope. Although the problem can be solved in polynomial time for rank lower than 3, the problem is NP-hard for input whose rank exceeds 3.

Simplicial Steinitz problem
The simplical Steinitz problem is a related algorithmic decision problem about polytopes. The problem is to determine if a given finite abstract polytope has a realization as a simplicial polytope. This problem is also NP-hard in the general case (non-fixed rank), and P for rank lower than 3, but is of unknown complexity in the fixed rank case.