There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the Ukrainian easter eggs by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2.
This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical
polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.
