Abstract
AbstractLetBbe ano-symmetric convex body in ℝd, andMdbe the normed space with unit ballB. TheMd-thickness ΔB(K) of a convex bodyK⊆ ℝdis the smallest possibleMd-distance between two distinct parallel supporting hyperplanes ofK. Furthermore,Kis said to beMd-reduced if ΔB(K′) < ΔB(K) for every convex bodyK′ withK′ ⊆KandK′ ≠K. In our main theorems we describeMd-reduced polytopes as polytopes whose face lattices possess certain antipodality properties. As one of the consequences, we obtain that if the boundary ofBis regular, then ad-polytope withmfacets andnvertices is notMd-reduced providedm=d+ 2 orn=d+ 2 orn>m. The latter statement yields a new partial answer to Lassak's question on the existence of Euclidean reducedd-polytopes ford≥ 3.
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