Separation of two convex sets in convexity structures
type: Item
instance of: scholarly article
title: Separation of two convex sets in convexity structures
zbMATH Open document ID: 0807.52002
DOI: 10.1007/BF01222661
author: Victor Chepoi
published in: Journal of Geometry
publication date: 1995-02-19T00:00:00Z
review text: 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.
reviewed by: Maria Moszyńska
Mathematics Subject Classification ID: 52A01
zbMATH DE Number: 621987
zbMATH Keywords: \(n\)-ary convexity
zbMATH Keywords: interval-convexity
zbMATH Keywords: separation
zbMATH Keywords: convexity structures
zbMATH Keywords: alignments
MaRDI profile type: Publication
cites work: Separation of two convex sets in convexity spaces and in straight line spaces
cites work: Graphs with intrinsic s3 convexities
cites work: A Helly theorem in weakly modular space
cites work: Median algebras
cites work: Pseudo-modular graphs
cites work: Q4873790
cites work: Convexity spaces. II: Separation
cites work: Isometric subgraphs of Hamming graphs and d-convexity
cites work: Distance-preserving subgraphs of hypercubes
cites work: Q4069907
cites work: On rigid circuit graphs
cites work: A general set-separation theorem
cites work: On local convexity in graphs
cites work: Ternary spaces, media, and Chebyshev sets
cites work: Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers
cites work: Subbases, convex sets, and hyperspaces
cites work: On the Helly property working as a compactness criterion on graphs
cites work: Collapse of the metric hierarchy for bipartite graphs
cites work: Conditions for invariance of set diameters under d-convexification in a graph
cites work: Q3310512
cites work: Binary Convexities and Distributive Lattices
cites work: Q3141898
description: scientific article; zbMATH DE number 621987