José Pedro Moreno

Structure of the Space of Diametrically Complete Sets in a Minkowski Space

We study the structure of the space of diametrically complete sets in a finite dimensional normed space. In contrast to the Euclidean case, this space is in general not convex. We show that its starshapedness is equivalent to the completeness of the parallel bodies of complete sets, a property studied in Moreno and Schneider (Isr. J. Math. 2012, doi:10.1007/s11856-012-0003-6), which is generically not satisfied. The space in question is, however, always contractible. Our main result states that in the case of a polyhedral norm, the space of translation classes of diametrically complete sets of given diameter is a finite polytopal complex. The proof makes use of a diagram technique, due to P. McMullen, for the representation of translation classes of polytopes with given normal vectors. The paper concludes with a study of the extreme diametrically complete sets.

Intersection properties of polyhedral norms

We investigate the familyM of intersections of balls in a finite dimensional vector space with a polyhedral norm. The spaces for whichM is closed under Minkowski addition are completely determined. We characterize also the polyhedral norms for which M is closed under adding a ball. A subset P ofM consists of the Mazur sets K, defined by the property that for any hyperplane H not meeting K there is a ball containing K and not meeting H. We characterize the Mazur sets in terms of their normal cones and also as summands of closed balls. As a consequence, we characterize the polyhedral spaces with only trivial Mazur sets as those whose unit ball is indecomposable.