Realization space of arrangements of convex bodies

In this talk I introduce combinatorial types of arrangements of convex bodies, an extension of order types of point sets to arrangements of convex bodies, and present some recent results about their realization spaces. Our results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. On one hand, we show that every combinatorial type can be realized by an arrangement of convex bodies and (under mild assumptions) its realization space is contractible. On the other, we prove a universality theorem that says that the restriction of the realization space to arrangements of convex polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.