Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called 'directed sets' is constructed, in which the convex cone of all nonempty convex compact sets in ℝⁿ is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n = 1. The directed sets in ℝⁿ are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a 'support' function and directed 'supporting faces' of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the 'support' function and recursively on the directed 'supporting faces'. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.