A convex body in Euclidean d-dimensional space is said to be reduced if it
has no proper convex subset with the same minimal width. It is not known
whether there are reduced polytopes for d > 2. The notion of reducedness
was also extended to Minkowski spaces, and in the talk recent results in
this direction will be presented. These results are also connected with
antipodality properties, more general boundary properties, and
cross-section measures of convex (bodies and) polytopes. For example, the
following surprising affine result and its relation to reducedness are
shown: no d-dimensional convex polytope with n vertices is strictly
vertex-facet antipodal if and only if n = d+2.

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