Separation of two convex sets in convexity structures
The well known notion of convex structure is a generalization of the classical notion of convexity in affine spaces: Let \(X\) be a nonempty set and \({\mathcal C}\) a collection of its subsets. The pair \((X, {\mathcal C})\) is a convex structure (called also a convexity structure) provided that \(\emptyset, X \in {\mathcal C}\) and \({\mathcal C}\) is closed under intersection of arbitrary subfamilies. The paper presents a survey of various classes of convexity structures, such as alignments, \(n\)-ary convexities, interval convexities. Its main purpose is to characterize \(S_ 4\)-convexity structures, i.e., structures satisfying the following axiom \(S_ 4\): any two disjoint sets \(A,B \in {\mathcal C}\) can be separated by a half-space. (A set \(H\) is a half-space provided that both \(H\) and \(X \backslash H\) belong to \(C)\). The abstract is unclear.
scientific article; zbMATH DE number 621987
scientific article