  It is well known that in $n$-dimensional Euclidean space ($n\geq 2$) 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 $n\geq 3$. Thus, the question occurs whether for $n\geq 3$ any
complete set is reduced. Answering this in the negative for $n\geq 3$, we
construct $(2^{k}-1)$-dimensional ($k\geq 2$) complete sets which are not
reduced.
