Nerves of convex sets: classification for remaining parameters

Classify the complexity of recognizing whether a given simplicial complex is the k‑skeleton of the nerve of c‑dimensional convex sets in ℝ^d for the parameter combinations not covered by current results.

Background

The nerve realizability problem depends on three parameters (k,c,d). For several ranges (e.g., k ≥ d, c ∈ {d−1,d}), ER-completeness is known. Some combinations are always realizable, making their recognition trivial.

The authors explicitly note that the remaining parameter regimes have not been settled.