Realization Spaces

The realization space of a combinatorial polytope is the set of all geometric polytopes that realize it. Mnëv's celebrated universality theorem states that realization spaces of polytopes can be arbitrarily complicated (stably equivalent to any primary basic semialgebraic set \(S\)). An outstanding improvement by Richter-Gebert shows that this holds already for \(4\)-dimensional polytopes. Moreover, when \(S\) is open, universality also holds for simplicial generic polytopes without restriction on the dimension as it was showed recently.

An emergent topic is the study of the space of all realizations of a fan (this is called the type cone of the fan), that is, the subset of the realization space containing all polytopes with a fixed normal fan. For the Coxeter arrangement, this is the cone of generalized permutahedra. It turns out that these realization spaces for the Cambrian fans (in connection to generalized associahedra) are simplicial, which has interesting applications in connection to theoretical physics.