Ordered incidence geometry and the geometric foundations of convexity theory (Q1093892)

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.