DIRECTED DERIVATIVES OF CONVEX COMPACT-VALUED MAPPINGS

Convex compact sets can be embedded into the Banach space of directed sets. Directed sets allow a visualization as possibly non-convex, compact sets in (cid:0)(cid:2)(cid:1) and hence, this space could be used to visualize differences of embedded convex compact sets. The main application is the visualization as well as the theoretical and numerical calculation of set-valued derivatives. Known notions of afﬁne, semi-afﬁne and quasi-afﬁne maps and their derivatives are studied.

, the Minkowski addition, the pointwise negative (multiplication by ) ) and the algebraic difference are defined as in [4,10] resp. [11] are convex, compact sets. is only satisfied in general. The last two differences have close connections to the visualization of differences of embedded convex compact sets (see (1.6)).

DIRECTED SETS
Most of the definitions presented here are recursive and recalled from [1,2].
The definition is motivated by the fact that the supporting face of each convex compact set lies on the hyperplane given by the support function in this direction. To enable a recursive approach, the support function is saved separately from the supporting face and the latter is seen as a § © ( ) -dimensional set.

Definition 2.2 The embedding
The embedding % ¤ is an isometric, injective map from which is equivalent to Demyanov's metric.
The operations introduced on are defined recursively and act separately on both components of the directed set.

Definition 2.3 Consider
the first components of the directed sets do not exist) as The space has remarkable properties with these operations which were studied in [2]. E.g., is a Banach space. Since we are interested in visualizations of differences of embedded elements of " § © £ ¤ , we restrict our attention to the closed linear hull of

Proposition 2.4 ([2]) The subspace
The following result proven in [2] shows that the well-known and often used operations on " § © £ ¤ commute with the embedding.

Proposition 2.5 ([2]) Consider
. Then, The visualization for a directed set , and the indefinite case, i.e. is a difference of two embedded convex compact sets or the limit of such a sequence. Therefore, the visualization splits in three parts, the convex part, the concave part and the mixed-type part. In the definition of the mixed-type part, the boundary part is involved.
The projection involved in Definition 2.2 splits into an orthonormal, linear part ¤ " ! # and a translation and allows an inverse reprojection on § The mixed-type part ¤ and the boundary part There is no mixed-type part for directed intervals, this part could appear only for dimensions ( r e 6 . The convex and the concave part are both convex, compact sets, but could be empty as in one example in [3]. The Let us collect special properties of the visualization parts and the boundary part which are proven in [3]. Especially, the boundary part and the visualization are never empty and the three parts of the visualization in Definition 2.6 are disjoint, except in the special case that the convex and the concave part equal to the same point. Theorem 2.8 states that the visualized difference of two embedded convex compacts lies between the geometric and the Demyanov difference (compare (1.2)).

DERIVATIVES OF SET-VALUED MAPPINGS
In this section, notions of simple convex-valued mappings are recalled from [8,12]. In the following,  , is quasi-affine with images in Thus, positive parts of special linear directed maps are quasi-affine maps.
Eclipsing maps introduced in [8] could be discontinuous (see [12]), so that directed differentiability for every eclipsing map could not be expected. . The numerical calculation of the derivative with difference quotients is neither very reliable nor very efficient and is chosen here only for simplicity of the presentation.   Proposition 3.6). In this case, the derivative is the inverse of the embedded Figure 1.1, so that the visualization is simply with inner normals (see Theorem 2.8 and Figure 1.2).  with outer normals and a nonempty, non-convex mixed-type part appears (see Theorem 2.8 and Figure 1.3).

SUMMARY
Embedded affine and semi-affine maps are linear directed maps. The situation for quasi-affine maps is not so easy, but specially chosen linear directed maps have positive parts which, seen as a convex-valued maps, are quasi-affine.
The concept of linear directed maps tries to unify the different notions of simple mappings with convex, compact images in one hand and allows on the other hand to study their differences by visualizing their derivative. Directed sets offer a convenient tool for studying the possible directed differentiability of set-valued mappings with the help of the visualization of difference quotients.