Although there is an abundance of abstract convexity theories, there seems to be no satisfying concept of convexity which is based only on Hilbert’s axioms of incidence and order. Not appealing to the projective embedding theorems for ordered spaces of dimension \(\geq 3\) [cf. say, K. Sörensen, Projektive Einbettung angeordneter Räume, TU München, Beiträge zur Geometrie und Algebra, 15, 8-35 (1986)] the author presents an abstract convexity theory for ordered spaces, which includes the planar case and may be regarded as a way between the classical theory in real space and the abstract theory of Bryant and Webster.
A first chapter contains a readable, concise stand alone introduction to the (mainly well known) fundamental notions and concepts of the theory of ordered spaces. Based on Hilbert’s axioms for orderings (slightly modified), it covers the exchange property of ordered spaces and the associated dimension theory, betweeness functions in the sense of Sperner, (relative) hyperplanes and their associated half spaces, and finally linear orderings on lines and on hyperplane bundles.
The main chapter introduces the notions of convexity and of convex hull in ordered spaces. A concept of local spaces allows to identify (relatively) intern and extreme points, components and faces of convex and compact like convex sets. In particular, the theorem of Carathéodory stating that any \(x \in \text{conv} (M)\) lies in the convex hull of a simplex with vertices in \(M\) goes over to ordered spaces. A discussion of convexity in hyperplanes and of intersections of hyperplanes with polytopes culminates in the concept of support hyperplanes: Each \(d\)-polytope in an ordered space of rank \(d+1\) is an intersection of finitely many closed supporting half spaces. Further, three natural topologies arising from the notions of convexity are exhibited. Their basic sets are (1) the intersections of finitely many open half spaces, (2) the point sets \(E\) with \(\text{int} (E)=E\), and (3) the convex point sets \(E\) with \(\text{int} (E)=E\). For the latter topology the interrelation between convex hull and topological closure is studied. The paper closes with a brief discussion of general half spaces, extending some classical results also of this topic to arbitrary ordered spaces.