Families of convex sets having convex union
We characterize the f-vectors of finite families of closed convex sets in \(R^ d\) whose union is also convex. (The f-vector of a family of sets is the sequence \((f_ 0,f_ 1,f_ 2,....)\), where \(f_ k\) denotes the number of intersecting subfamilies of size \(k+1.)\) The characterization is by a system of linear and nonlinear inequalities (and one equality) involving certain linear combinations of the \(f_ k's\). The proof relies on the following facts: 1) The f-vectors of arbitrary families of convex sets in \(R^ d\) have been determined by \textit{G. Kalai} [Isr. J. Math. 48, 175-195 (1984; Zbl 0572.52006)]. 2) The nerve of a finite family of closed convex sets having convex union is acyclic. This is a special case of a theorem of \textit{J. Leray} [J. Math. Pur. Appl., IX. Ser. 24, 95-167 (1945; Zbl 0060.407)]. 3) For each acyclic simplicial complex there exists a cone complex with the same f-vector. (The f-vector of a complex is the sequence \((f_ 0,f_ 1,f_ 2,....)\), where \(f_ k\) denotes the number of k- dimensional faces.) This is again due to \textit{G. Kalai} [Discrete Math. 55, 97-99 (1985; Zbl 0579.57015)]. 4) If an acyclic complex is d-collapsible in the sense of \textit{G. Wegner} [Arch. Math. 26, 317-321 (1975; Zbl 0308.52005)], then the cone in 3) can be chosen to be d-collapsible.
finite family of closed convex sets having convex union
scientific article; zbMATH DE number 3984011
scientific article