Geodesic
Contractibility of Half-Spaces of Partial Convexity
Matematičeskie zametki, Tome 85 (2009) no. 6, pp. 915-926.

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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.
Keywords: partial convexity, orthoconvexity, half-space of partial convexity, directed half-space, Fink–Wood problem. 
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V. G. Naidenko. Contractibility of Half-Spaces of Partial Convexity. Matematičeskie zametki, Tome 85 (2009) no. 6, pp. 915-926. http://geodesic.mathdoc.fr/item/MZM_2009_85_6_a8/

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