Faces of convex sets and Minkowski additive selections (Q1345513)

Consider the set \({\mathcal K}^n\) of all convex bodies in \(\mathbb{R}^n\) and a convex cone \({\mathcal F} \subset {\mathcal K}^n\) which contains all segments. A Minkowski additive mapping \(T : {\mathcal F} \to {\mathcal K}^n\) is called a face mapping if \(T(K)\) is a face of \(K\), for all \(K \in {\mathcal F}\). The main result of the paper gives a complete characterization of \(\Phi_k ({\mathcal F})\), the set of face mappings \(T\) on \({\mathcal F}\) of order \(k\) (this means \(\dim T(K) \leq k\) for all \(K \in {\mathcal F}\), and \(\dim T(K) = k\) for at least one \(K \in {\mathcal F})\). It is shown that all elements of \(\Phi_k ({\mathcal F})\) are obtained from \((n - k)\)-frames (ordered \((n - k)\)-tuples of orthonormal vectors \(x_1, \ldots, x_{n - k})\) by taking subsequent support sets of convex bodies \(K\) (the support set \(K(x_1)\) of \(K\) in direction \(x_1\), the support set of \(K(x_1)\) in direction \(x_2\), etc.). Further results concern the set \(\Phi_0 ({\mathcal F})\) which may be viewed as a set of additive selections on \({\mathcal F}\) (in fact, \(\Phi_0 ({\mathcal F})\) is a subset of the class \({\mathcal E} ({\mathcal F})\) of extremal selections). It is shown how the closure of \(\Phi_0 ({\mathcal F})\) depends on the structure of \({\mathcal F}\). Finally, the sets \(\Phi_0 ({\mathcal F})\) and \({\mathcal E} ({\mathcal F})\) are described in the case where \({\mathcal F}\) is the class of zonotopes in \(\mathbb{R}^n\).