On a conjecture of Fink and Wood (Q1039858)

The author considers the so-called \(\mathcal O\)-convexity, see the paper [J. Geom. 62, No.~1--2, 99--120 (1998; Zbl 0923.52001)] by \textit{E. Fink} and \textit{D. Wood}, who asked if arbitrary connected \(\mathcal O\)-half-space is contractible provided the orientation set \(\mathcal O\) has the point-intersection property. The present note shows that this is true in every two-dimensional space but not true in some three-dimensional spaces.