On reduced polytopes and antipodality

Abstract

Abstract Let B be an o-symmetric convex body in ℝ d , and Md be the normed space with unit ball B. The Md -thickness Δ B (K) of a convex body K ⊆ ℝ d is the smallest possible Md -distance between two distinct parallel supporting hyperplanes of K. Furthermore, K is said to be Md -reduced if Δ B (K′) < Δ B (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.