On Reduced Polytopes and Antipodality

Let B be an o-symmetric convex body in R, and M be the normed space with unit ball B. The M-thickness ∆B(K) of a convex body K ⊆ R is the smallest possible Mdistance between two distinct parallel supporting hyperplanes of K. Furthermore, K is said to be M-reduced if ∆B(K ′) < ∆B(K) for every convex body K ′ with K ′ ⊆ K and K ′ 6= K. In our main theorems we describe M-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-reduced for 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. 2000 Mathematics Subject Classification. Primary 52A20, 52B12; Secondary 52A21, 46B20