  We derive tight expressions for the maximum number of $k$-faces,
$0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex
$d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a
(recursively defined) function on the number of vertices of the polytopes.
  Our results coincide with those recently proved by Adiprasito and Sanyal [2].
In contrast to Adiprasito and Sanyal's approach, which uses tools from
Combinatorial Commutative Algebra, our approach is purely geometric and uses
basic notions such as $f$- and $h$-vector calculus and shellings, and
generalizes the methodology used in [15] and [14] for proving upper bounds on
the $f$-vector of the Minkowski sum of two and three convex polytopes,
respectively.
  The key idea behind our approach is to express the Minkowski sum
$P_1+...+P_r$ as a section of the Cayley polytope $\mathcal{C}$ of the
summands; bounding the $k$-faces of $P_1+...+P_r$ reduces to bounding the
subset of the $(k+r-1)$-faces of $\mathcal{C}$ that contain vertices from each
of the $r$ polytopes.
  We end our paper with a sketch of an explicit construction that establishes
the tightness of the upper bounds.

Computational Geometry;Combinatorics