Nerves of Convex Sets: Remaining Parameter Regimes

Classify the computational complexity of recognizing k-skeletons of nerves of c-dimensional convex sets in R^d for all parameter combinations not covered by known results.

Background

The nerve recognition problem’s complexity depends on three parameters: the simplicial complex dimension k, the convex set dimension c, and the ambient dimension d.

The survey summarizes several ER-complete and polynomial-time regimes and explicitly states that the remaining parameter combinations are open.