  Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$
and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$,
$1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of
$k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by
$\Phi_{k+r}(n_1,n_2,...,n_r)$, where $\Phi_{\ell}(n_1,n_2,...,n_r)=
\sum_{\substack{1\le{}s_i\le{}n_i s_1+...+s_r=\ell}}
\prod_{i=1}^r\binom{n_i}{s_i}$, $\ell\ge{}r$. Fukuda and Weibel have also shown
that the upper bound mentioned above is tight for $d\ge{}4$,
$2\le{}r\le{}\lfloor\frac{d}{2}\rfloor$, and for all
$0\le{}k\le{}\lfloor\frac{d}{2}\rfloor-r$.
  In this paper we construct a set of $r$ neighborly $d$-polytopes
$P_1,P_2,...,P_r$, where $d\ge{}3$ and $2\le{}r\le{}d-1$, for which the upper
bound of Fukuda and Weibel is attained for all
$0\le{}k\le{}\lfloor\frac{d+r-1}{2}\rfloor-r$. Our approach is based on what is
known as the Cayley trick for Minkowski sums. A direct consequence of our
result is a tight asymptotic bound on the complexity of the Minkowski sum
$P_1+P_2+...+P_r$, for any fixed dimension $d$ and any $2\le{}r\le{}d-1$, when
the number of vertices of the polytopes is (asymptotically) the same.
