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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 fromCombinatorial Commutative Algebra, our approach is purely geometric and usesbasic notions such as $f$- and $h$-vector calculus and shellings, andgeneralizes the methodology used in [15] and [14] for proving upper bounds onthe $f$-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 thesummands; bounding the $k$-faces of $P_1+...+P_r$ reduces to bounding thesubset of the $(k+r-1)$-faces of $\mathcal{C}$ that contain vertices from eachof the $r$ polytopes.  We end our paper with a sketch of an explicit construction that establishesthe tightness of the upper bounds.

A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes