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 orthocon-vexity 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.