We begin with some basic notions and results of convex geometry that we need, as polyhedra, cones, spherical polyhedra, hyperplane arrangements and CW-complexes. In Chapter 3 we introduce convex n-partitions and we prove that all the regions of a partition must be polyhedral. Then we define some related notions, such as spherical partitions and the face structure, and prove some basic facts about them.
In Chapter 4 we look at the space C(R^d, n) of all convex n-partitions of R^d, describing the metric structure there that fixes the topology of the space and also a natural compactification C(R^d, ≤n) where empty regions are allowed. Then we prove that spaces of
n-partitions are union of semialgebraic pieces in two different ways. We look at hyperplane arrangements carrying an n-partition, and give a description of C(R^d, n) where the pieces depend on the hyperplanes used to obtain the partition (Theorem 4.14). For the
second description we need to introduce nodes and node systems that are a generalization of the vertices, and define the combinatorial type of a partition. These combinatorial types give the semialgebraic pieces that build the spaces (See Theorem 4.47). At the end of the chapter we describe explicitly particular spaces of n-partitions of R^d and their compactifications for n = 2 and also for d = 1.
In Chapter 5 we talk about regular partitions and mention some known results about them. Using these results we compute the dimension of the space of regular partitions Creg(R^d, n). Then we prove a universality theorem that says that realization spaces
of regular partitions can be stably equivalent to any primary basic semialgebraic set.
In Chapter 6 we investigate the dimensions of realization spaces. We first study the case d = 2 and find that for large n the dimension of C(R^2, n) is much bigger than dim(Creg(R^2, n)). Then we focus on the case d = 3, where we conjecture that the dimension of
C(R^3, n) is equal to the dimension of Creg(R^3, n) and try to justify this with a heuristic counting for the dimension of each realization space. From this counting we find an incidence theorem for 3-polytopes and find many examples of partitions where this counting works.
In Chapter 7 we introduce the spaces of equipartitions
Cequi(R^d, n, µ) given a positive bounded measure µ. We explore the topological structure of some small cases of spaces of
equipartitions and using this, we describe the spaces of n-partitions for d = 2 and n = 3. We also discuss the Nandakumar and Ramana Rao problem and different equivariant maps that show that considering regular equipartitions is as good as considering all
equipartitions with respect to the approach based on configuration spaces to find fair partitions. We end by listing some further questions that for now remain open.