Local topology of the free complex of a two-dimensional generalized convex shelling (Q932635)

A generalized convex shelling was introduced by \textit{K. Kashibawara}, \textit{M. Nakamura} and \textit{Y. Okamoto} [Comput. Geom. 30, No. 2, 129--144 (2005; Zbl 1113.52002)] for their representation theorem of convex geometries in the sense of \textit{P. H. Edelman} and \textit{R. E. Jamison} [Geom. Dedicata 19, 247--270 (1985; Zbl 0577.52001)]. Given two finite sets \(P\), \(Q\) in the real plane a generalized convex shelling on \(P\) with respect to \(Q\) is considered. Its free sets form a simplicial complex \(F\). It is shown that the deletion of one point from \(F\) is either contractible or homotopy equivalent to two distinct points.