Contractibility of half-spaces of partial convexity
The author of this paper deals with the following Fink-Wood conjecture: every connected half-space of partial convexity (convexity with respect to a set of directions, or of directing hyperplanes) is contractible in \(\mathbb{R}^n\) if the set of directing hyperplanes has the point intersection property. A counterexample is constructed in the three-dimensional space, proving that there is a connected non-simply-connected half-space of orthoconvexity not verifying the Fink-Wood conjecture. However, it is shown that this conjecture stays valid in case of directed half-spaces of partial convexity. Thus, the main result presented in this paper is: every directed half-space \(X\) of partial convexity, in the linear \(n\)-dimensional space, is contractible provided that the set of directing hyperplanes has the point intersecting property.
scientific article; zbMATH DE number 5628218