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.
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© 1997 Springer-Verlag Berlin Heidelberg
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Boltyanski, V., Martini, H., Soltan, P.S. (1997). d-Convexity in normed spaces. In: Excursions into Combinatorial Geometry. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59237-9_2
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DOI: https://doi.org/10.1007/978-3-642-59237-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61341-1
Online ISBN: 978-3-642-59237-9
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