Complete sets need not be reduced in Minkowski spaces (Q747572)

The diameter of a convex body in a Minkowski space may be defined as the maximum distance between any pair of parallel supporting hyperplanes. The body is called diametrally complete if every proper superset has strictly greater diameter. Likewise, the thickness of the body is defined as the minimum distance between any pair of parallel supporting hyperplanes; and the body is called reduced if every proper subset has strictly smaller thickness. In a Euclidean space, every complete convex body has constant width and therefore is reduced. A reasonable conjecture would be that every complete convex body in a Minkowski space is also reduced; this turns out to be be false. In this paper Minkowski spaces of arbitrarily high dimension are exhibited, in each of which some simplex is diametrally complete set but not reduced.