  Studying the geometry of sets appearing in various problems of quantum
information helps in understanding different parts of the theory. It is thus
worthwhile to approach quantum mechanics from the angle of geometry -- this has
already provided a multitude of interesting results. In this thesis I
demonstrate results relevant to numerical ranges -- the sets of simultaneously
attainable expectation values of several observables. In particular, I apply
this notion in the problems related to uncertainty relations, entanglement
detection, and determining bounds for the value of spectral gap. Apart from
this, I present geometric structures helping with the question of state
interconversion using channels commuting with a particular representation of a
group.
