Total Lattices of Convex Sets and of Linear Spaces

Abstract

Linear spaces and convex sets are two of the most important and familiar classes of geometric object. Their general importance in modern mathematics would seem to merit an autonomous treatment of each of these notions. For the set of linear spaces of a finite dimensional projective this has been given by Garrett Birkhoff [1] 2in his characterization of this set as a lattice, which may be considered to be an abstract theory of linearity based on the notions linear space and inclusion. Menger [8, 9] also has solved this problem and has characterized the linear space lattice of a finite dimensional affine geometry. We consider the corresponding problem for the notion convexity, namely, to characterize the lattice of all convex sets of an appropriate space.3 At the outset we are confronted with the problem of selecting the domain from which the convex sets are to be chosen, for example-Euclidean, hyperbolic or affine geometry, a real linear vector space or the interior of an n-dimensional simplex. It would seem that the underlying should include all these examples and should be of the most general type in which convex sets exist and have their usual descriptive (that is, nonmetrical) properties. Since may be defined as closure under the operation of joining points by segments, the we seek should be a general type of linear based on a notion of intermediacy4 like segment or betweenness. Such an ordered linear geometry is well known in the foundations of under the name descriptive geometry. As ordinarily developed the foundational aspect of the subject has completely dominated its treatment, which is a detailed but rather atomistic study of segments, rays, lines, triangles etc.that is of the simplest convex sets. On the other hand, we may conceive of descriptive as a general theory of convex sets arising by abstraction from the convexity properties of the spaces mentioned in the last paragraph. This is the viewpoint which we shall adopt. Thus our problem may be stated precisely as follows: to characterize the lattice of convex sets of a descriptive of arbitrary dimension, finite or infinite. In the course of the investigation there became evident certain close similarities between this lattice and the lattice of linear spaces of a projective which enabled us to characterize both types of lattices simultaneously, distinguishing between them by the assertion or denial of a single property. The basis