Realization spaces of arrangements of convex bodies

TL;DR

This paper extends the concept of order types to arrangements of convex bodies, studying their realization spaces and revealing a trade-off between combinatorial and topological complexities, including universality results.

Contribution

It introduces combinatorial types for convex body arrangements and proves universality theorems relating their realization spaces to arbitrary semialgebraic sets.

Findings

Every combinatorial type is realizable with a contractible realization space.

The realization space of arrangements of polygons with bounded vertices can have any homotopy type.

The study establishes a fundamental link between combinatorial complexity and topological properties.

Abstract

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.