Ordered incidence geometry and the geometric foundations of convexity theory
type: Item
instance of: scholarly article
title: Ordered incidence geometry and the geometric foundations of convexity theory
zbMATH Open document ID: 0629.51013
DOI: 10.1007/BF01227810
author: Aharon Ben-Tal
author: Adi Ben-Israel
published in: Journal of Geometry
publication date: 1987-01-01T00:00:00Z
review text: The authors introduce the concept of an Ordered Incidence Geometry (OIG), which provides a setting for the study of convexity from an axiomatic viewpoint. An OIG is a triple \(\{\) X,\({\mathcal A},\dim \}\), where X is the space of ``points'', \({\mathcal A}\) is the family of ``affine sets'' and dim: \({\mathcal A}\to Z\cup \{-\infty \}\) is the ``dimension'' function, together with a ternary relation (betweeness) on X and ten axioms, pertaining to the following groups: 1) \({\mathcal A}\) is closed under arbitrary intersections and contains X, \(\emptyset\) and all singletons; 2) monotonicity properties of dimension; 3) axioms concerning the relation betweeness. In contrast with other existing axiomatics for convexity, the basic notion here is that of affine set, as in the Hilbert's approach to the foundations of geometry. Separation theorems, as well as those of Radon and Helly, are proved to hold for OIG. An application to generalized convex functions defined on an open real interval is given.
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 4024146
zbMATH Keywords: convexity geometry
zbMATH Keywords: axiomatic convexity
zbMATH Keywords: Fenchel duality
zbMATH Keywords: Radon theorem
zbMATH Keywords: Helly theorem
zbMATH Keywords: Ordered Incidence Geometry
zbMATH Keywords: betweeness
zbMATH Keywords: Separation theorems
zbMATH Keywords: generalized convex functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4024146