Convex, compact, nonempty sets can be embedded in the Banach space D(n) of directed sets. A directed set is parametrized by unit vectors l Sn-1 and consists of two components, a uniformily bounded function with (n - 1)-dimensional directed sets and a continuous function an(.). Recursively defined operations act separately on the two components and generalize the Minkowski addition and scalar multiplication for convex sets. One-dimensional directed sets are equivalent to generalized intervals.
Directed sets allow visualizations as possibly non-convex, compact sets in n, being defined as a union of three parts, the convex part, the concave part (both are convex sets) and the (non-convex) mixed-type part. The three parts together with orientation bundles represent the three main cases: either the directed set is an embedded convex compact set C (outer normals, the visualization equals the convex part C), the directed set is an inverse of such an element (inner normals, the visualization equals the concave part, the pointwise inverse of C), or the directed set is a difference of two embedded elements or a limit of such a sequence. In the last case, the mixed-type part is not empty.
As a main application the theoretical study and visualization of derivatives of embedded simple maps with convex, compact images are presented. Affine and semi-affine maps lead to so-called linear directed maps with constant derivatives. Quasi-affine maps can be constructed by calculating the convex part of linear directed maps.