  It is well known that correlations predicted by quantum mechanics cannot be
explained by any classical (local-realistic) theory. The relative strength of
quantum and classical correlations is usually studied in the context of Bell
inequalities, but this tells us little about the geometry of the quantum set of
correlations. In other words, we do not have good intuition about what the
quantum set actually looks like. In this paper we study the geometry of the
quantum set using standard tools from convex geometry. We find explicit
examples of rather counter-intuitive features in the simplest non-trivial Bell
scenario (two parties, two inputs and two outputs) and illustrate them using
2-dimensional slice plots. We also show that even more complex features appear
in Bell scenarios with more inputs or more parties. Finally, we discuss the
limitations that the geometry of the quantum set imposes on the task of
self-testing.
