Complete sets need not be reduced in Minkowski spaces

It is well known that in $n$-dimensional Euclidean space ($n\ge 2$) the classes of (diametrically) complete sets and of bodies of constant width coincide. Due to this, they both form a proper subfamily of the class of reduced bodies. For $n$-dimensional Minkowski spaces, this coincidence is no longer true if $n\ge 3$. Thus, the question occurs whether for $n\ge 3$ any complete set is reduced. Answering this in the negative for $n\ge 3$, we construct $(2^k-1)$-dimensional ($k\ge 2$) complete sets which are not reduced.