d-Convexity in normed spaces

Abstract

In this chapter we consider basic properties of d-convex sets in finite-dimensional normed spaces. Central subjects are the support properties of d-convex sets (§10) and the properties of d-convex flats (§11). This chapter is of decisive importance for developing a machinery for solving combinatorial problems. Nevertheless, it is interesting for itself, because the family of d-convex sets has far-reaching analogies to the family of convex sets in Rⁿ. For example, the carrying flats, faces, inscribed cones, and supporting cones of d-convex sets are d-convex themselves; moreover, each boundary point of a d-convex body is contained in a d-convex supporting hyperplane. Questions referring to separability of d-convex sets are considered in section 13 of this chapter.