Collapsibility of the k-flat intersection complex

Establish whether, for every finite family F of convex sets in R^d each containing a flat of dimension at least k, the simplicial complex N_k(F)—whose vertices correspond to the sets in F and whose faces are precisely the subfamilies whose intersection contains a k-flat—is (d-k)-collapsible.

Background

The paper defines the simplicial complex N_k(F) for a finite family F of convex sets in R^d, with a face corresponding exactly to a subfamily whose intersection contains a k-flat. When k=0, this is the usual nerve complex N_0(F), which is known to be d-collapsible.

Proving that N_k(F) is (d-k)-collapsible would yield substantially stronger bounds for the constant β in the fractional De Santis theorem by allowing direct application of results of Kalai. It would also imply the selection-structure De Santis theorem as a direct consequence of a cited theorem. The authors state that the standard swiping arguments proving collapsibility for ordinary nerves do not appear to extend to N_k(F).