| Summary: | <p>We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$ in $\mathbb{R}^d$, as a function of the number of vertices of the polytopes, for any $d\ge{}2$.<br /> <br /> Expressing the Minkowski sum as a section of the Cayley polytope $\mathcal{C}$ of its summands, counting the $k$-faces of $P_1+P_2+P_3$ reduces to counting the $(k+2)$-faces of $\mathcal{C}$ that contain vertices from each of the three polytopes.<br /> <br /> In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes in $\mathbb{R}^d$, where $r\ge d$. For $d\ge{}4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves.</p>
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