Directed sets and differences of convex compact sets
Robert Baier, Elza M. Farkhi · 2022
A linear normed and partially ordered space is introduced, in which the convex cone of all nonempty convex compact sets is embedded. This space of so-called “directed sets” is a Banach and a Riesz space for dimension n ≤ 2 and a Banach lattice. The basic differences of the approach to other existing embeddings are that there are no equivalence classes and secondly, that differences of directed convex sets are not real-valued functions of n arguments, but higher-dimensional maps representable as oriented manifolds, e.g. oriented curves/surfaces in the cases n = 2,3. Each part of the boundary of the inverse of a directed set is the negative of the boundary part of the directed set itself.