Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps (i.e. maps, which can be extended to affine maps between the whole euclidean spaces). The Minkowski sum of two polytopes $P,Q \subseteq \mathbb{R}^d$ is defined as $P + Q := \{ p+q : p \in P, q \in Q \}$, which is again a polytope.
Is there any categorical description of $P+Q$ as limit or colimit in $\textbf{Poly}$?
Edit: As it seems that there is no such description in that general situation, let us change the category and consider only polytopes which are centered at the origin and linear maps. Does it change anything?