  We derive tight expressions for the maximum number of $k$-faces,
$0\le{}k\le{}d-1$, of the Minkowski sum, $P_1\oplus{}P_2$, of two
$d$-dimensional convex polytopes $P_1$ and $P_2$, as a function of the number
of vertices of the polytopes.
  For even dimensions $d\ge{}2$, the maximum values are attained when $P_1$ and
$P_2$ are cyclic $d$-polytopes with disjoint vertex sets. For odd dimensions
$d\ge{}3$, the maximum values are attained when $P_1$ and $P_2$ are
$\lfloor\frac{d}{2}\rfloor$-neighborly $d$-polytopes, whose vertex sets are
chosen appropriately from two distinct $d$-dimensional moment-like curves.
