Approximate Depth of Convex polytopes

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our main results imply that simplices can only be “trivially approximated”. On the way, we obtain a characterization of simplices as the only “outer additive” convex bodies. This translates to the inability to approximate functions with less depth in a certain model of computation using maximum-gate neural networks.