The Structure of Polytopes in an Ordered Geometry
This chapter falls into three parts. First the theory of extreme sets and extremal linear spaces for convex sets is employed to elicit basic facial structural properties of polytopes in ordered geometries. The chief result is that a nontrivial polytope can be represented as the intersection of a certain finite family of closed halfspaces. These halfspaces have edges which are linear hulls of the maximal proper faces, or facets, of the polytope. Secondly, conditions are obtained, involving intersection properties with lines and segments, for a convex set to be the convex hull of its extreme points. Thirdly two characterizations of polytopes as types of convex set are obtained. One is based on the intersection properties mentioned above, the other on the representation of a polytope as an intersection of closed half spaces. The second characterization yields the result that two polytopes which meet intersect in a polytope.
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