Abstract
Let B be an o-symmetric convex body in ℝd, and Md be the normed space with unit ball B. The Md-thickness ΔB(K) of a convex body K ⊆ ℝd is the smallest possible Md-distance between two distinct parallel supporting hyperplanes of K. Furthermore, K is said to be Md-reduced if ΔB(K′) < ΔB(K) for every convex body K′ with K′ ⊆ K and K′ ≠ K. In our main theorems we describe Md-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 Md-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.
© de Gruyter 2008