On reduced polytopes and antipodality

Let B be an o-symmetric convex body in R(d), and M(d) be the normed space with unit ball B. The M(d)-thickness Delta(B) (K) of a convex body K subset of R(d) is the smallest possible M(d)-distance between two distinct parallel supporting hyperplanes of K. Furthermore, K is said to be M(d)-reduced if Delta(B) (K') < Delta(B) (K) for every convex body K' with K' subset of K and K' not equal K. In our main theorems we describe M(d)-reduced polytopes as polytopes whose face lattices possess certain antipodality properties. As one of the consequences, we obtain that if the boundary of B is regular, then a d-polytope with m facets and n vertices is not M(d)-reduced provided m = d + 2 or n = d + 2 or n > m. The latter statement yields a new partial answer to Lassak's question on the existence of Euclidean reduced d-polytopes for d >= 3.