The Universality Theorems for Oriented Matroids and Polytopes

Universality Theorems are exciting achievements in the theories of polytopes and oriented matroids. This article surveys the main developments in that context. We explain the basic constructions that lead to Universality Theorems. In particular, we show that one can use the Universality Theorem for rank 3 oriented matroids to obtain a Universality Theorem for 6-dimensional polytopes. 1 Universality Theorems Oriented matroids and polytopes are most fundamental objects in combinatorial geometry. A major breakthrough in both fields was the development of so called Universality Theorems. Intuitively speaking, a universality theorem states that realization spaces of oriented matroids (resp. polytopes) can be very complicated objects. For any reasonable complexity measure (like algorithmic complexity, topological complexity, algebraic complexity, etc.) questions related to realization spaces are as difficult as the corresponding problem for general systems of polynomial inequalities. It is the purpose of this article to sketch the main developments and achievements in this field over the last decade, to clarify the main concepts and to give an idea of the proof techniques that were applied. In principle, this article could be used as a “quick reference guide” for the constructions (not for the proofs) that lead to the Universality Theorems for oriented matroids and polytopes. Supported by the DFG Gerhard-Hess-Forschungsförderungspreis awarded to G.M. Ziegler