On Reduced Bodies, Regularity of Norms, and Related Antipodality Properties

The thickness (or minimum width) of a convex body K in Euclidean space E, d ≥ 2, is the minimal distance between two distinct parallel supporting hyperplanes of K. The body K is said to be reduced if no proper convex subset of K has the same thickness. It is natural to study the notions of thickness and reduced body also in Minkowski spaces (that is, in real normed linear spaces of finite dimension). We will prove various basic results on reduced polytopes in a d-dimensional Minkowski spaceM, where the degree of regularity, that the boundary of the unit ball of M has, plays an essential role. If P is a reduced polytope in a Minkowski space M whose unit ball is m-regular, say, then every vertex of P is strictly antipodal to some (m+1)-face of P , and this statement can be extended to a characterization of reduced bodies. With the help of this general property we obtain the following further results: in a Minkowski space M with Csmooth (i.e., (d− 2)-regular in our terms) unit ball there are no reduced polytopes with d+2 vertices or d+2 facets, while any simple reduced polytope inM has to be a simplex; for d = 2 the unit ball of M is C-smooth if and only if no reduced quadrilateral exists; if the unit ball of M is C-smooth, then reduced polygons in M necessarily have an odd number of vertices. We present also new results (used as lemmas for the main statement) in the spirit of classical convexity. These refer to the boundary structure of general convex bodies and, in particular, to strict antipodality of faces of polytopes in E, but also to cross-section measures in Minkowski spaces. For example, we prove the following surprising affine result: no d-dimensional convex polytope with n vertices is strictly vertex-facet antipodal if and only if n = d + 2. Mathematics Subject Classification (AMS 2000): 52A21, 52A20, 52B12, 46B20