Contractibility of Half-Spaces of Partial Convexity

The Fink–Wood problem on the contractibility of half-spaces of partial convexity is studied. It is proved that there exists a connected non-simply-connected half-space of orthoconvexity in the three-dimensional space, which disproves the Fink–Wood conjecture in the general case. In a special case, it is proved that, if the set of directions of partial convexity contains a basis of the linear $n$-dimensional space, then all directed half-spaces of partial convexity are contractible.